Optical sensor utilizing hollow-core photonic bandgap fiber with low phase thermal constant

ABSTRACT

An optical sensor includes a directional coupler comprising at least a first port, a second port, and a third port. The first port is in optical communication with the second port and with the third port such that a first optical signal received by the first port is split into a second optical signal that propagates to the second port and a third optical signal that propagates to the third port. The optical sensor further includes a photonic bandgap fiber having a hollow core and an inner cladding generally surrounding the core. The photonic bandgap fiber is in optical communication with the second port and with the third port. The second optical signal and the third optical signal counterpropagate through the photonic bandgap fiber and return to the third port and the second port, respectively. The photonic bandgap fiber has a phase thermal constant S less than 8 parts-per-million per degree Celsius.

CLAIM OF PRIORITY

This application claims benefit under 35 U.S.C. §119(e) to U.S.Provisional Patent Application No. 60/817,514, filed Jun. 29, 2006 andU.S. Provisional Patent Application No. 60/837,891, filed Aug. 14, 2006,each of which is incorporated in its entirety by reference herein.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to fiber optic sensors, and moreparticularly, relates to fiber optic interferometers for sensing, forexample, rotation, movement, pressure, or other stimuli.

2. Description of the Related Art

A fiber optic Sagnac interferometer is an example of a fiber opticsensor that typically comprises a loop of optical fiber to whichlightwaves are coupled for propagation around the loop in oppositedirections. After traversing the loop, the counterpropagating waves arecombined so that they coherently interfere to form an optical outputsignal. The intensity of this optical output signal varies as a functionof the relative phase of the counterpropagating waves when the waves arecombined.

Sagnac interferometers have proven particularly useful for rotationsensing (e.g., gyroscopes). Rotation of the loop about the loop'scentral axis of symmetry creates a relative phase difference between thecounterpropagating waves in accordance with the well-known Sagnaceffect, with the amount of phase difference proportional to the looprotation rate. The optical output signal produced by the interference ofthe combined counterpropagating waves varies in power as a function ofthe rotation rate of the loop. Rotation sensing is accomplished bydetection of this optical output signal.

Rotation sensing accuracies of Sagnac interferometers are affected byspurious waves caused by Rayleigh backscattering. Rayleigh scatteringoccurs in present state-of-the-art optical fibers because the smallelemental particles that make up the fiber material cause scattering ofsmall amounts of light. As a result of Rayleigh scattering, light isscattered in all directions. Light that is scattered forward and withinthe acceptance angle of the fiber is the forward-scattered light. Lightthat is scattered backward and within the acceptance angle of the fiberis the back-scattered light. In a fiber-optic gyroscope (FOG), both theclockwise and the counterclockwise waves along the sensing coil(referred to here as the primary clockwise and primary counterclockwisewaves) are scattered by Rayleigh scattering. The primary clockwise waveand the primary counterclockwise wave are both scattered in respectiveforward and backward directions. This scattered light returns to thedetector and adds noise to the primary clockwise wave and to thesecondary counterclockwise wave. The scattered light is divided into twotypes, coherent and incoherent. Coherently scattered light originatesfrom scattering occurring along the section of fiber of length L_(c)centered around the mid-point of the coil, where L_(c) is the coherencelength of the light source. This scattered light is coherent with theprimary wave from which it is derived and interferes coherently with theprimary wave. As a result, a sizeable amount of phase noise is produced.Forward coherent scattering is in phase with the primary wave from whichit is scattered, so it does not add phase noise. Instead, this forwardcoherent scattering adds shot noise. The scattered power is so smallcompared to the primary wave power that this shot noise is negligible.All other portions of the coil produce scattered light that isincoherent with the primary waves. The forward propagating incoherentscattered light adds only shot noise to the respective primary wave fromwhich it originates, and this additional shot noise is also negligible.The dominant scattered noise is coherent backscattering. This coherentbackscattering noise can be large. The coherent backscattering noise hasbeen reduced historically by using a broadband source, which has a veryshort coherence length L_(c). With a broadband source, the portion ofbackscattering wave originates from a very small section of fiber,namely a length L_(c) of typically a few tens of microns centered on themid-point of the fiber coil, and it is thus dramatically reducedcompared to what it would be with a traditional narrowband laser, whichhas a coherence length upward of many meters. See for example, HervéLefèvre, The Fiber-Optic Gyroscope, Section 4.2, Artech House, Boston,London, 1993, and references cited therein.

Rotation sensing accuracies are also affected by the AC Kerr effect,which cause phase differences between counterpropagating waves in theinterferometers. The AC Kerr effect is a well-known nonlinear opticalphenomena in which the refractive index of a substance changes when thesubstance is placed in a varying electric field. In optical fibers, theelectric fields of lightwaves propagating in the optical fiber canchange the refractive index of the fiber in accordance with the Kerreffect. Since the propagation constant of each of the waves traveling inthe fiber is a function of refractive index, the Kerr effect manifestsitself as intensity-dependent perturbations of the propagationconstants. If the power circulating in the clockwise direction in thecoil is not exactly the same as the power circulating in thecounterclockwise direction in the coil, as occurs for example if thecoupling ratio of the coupler that produces the two counterpropagatingwaves is not 50%, the optical Kerr effect will generally cause the wavesto propagate with different velocities, resulting in anon-rotationally-induced phase difference between the waves, and therebycreating a spurious signal. See, for example, pages 101-106 of theabove-cited Hervé Lefèvre, The Fiber Optic Gyroscope, and referencescited therein. The spurious signal is indistinguishable from arotationally induced signal. Fused silica optical fibers exhibitsufficiently strong Kerr nonlinearity that for the typical level ofoptical power traveling in a fiber optic gyroscope coil, theKerr-induced phase difference in the fiber optic rotation sensor may bemuch larger than the phase difference due to the Sagnac effect at smallrotation rates.

Silica in silica-based fibers also can be affected by magnetic fields.In particular, silica exhibits magneto-optic properties. As a result ofthe magneto-optic Faraday effect in the optical fiber, a longitudinalmagnetic field of magnitude B modifies the phase of a circularlypolarized wave by an amount proportional to B. The change in phase ofthe circularly polarized wave is also proportional to the Verdetconstant ν of the fiber material and the length of fiber L over whichthe field is applied. The sign of the phase shift depends on whether thelight is left-hand or right-hand circularly polarized. The sign alsodepends on the relative direction of the magnetic field and the lightpropagation. As a result, in the case of a linearly polarized light,this effect manifests itself as a change in the orientation of thepolarization by an angle θ=VBL. This effect is non-reciprocal. Forexample, in a Sagnac interferometer or in a ring interferometer whereidentical circularly polarized waves counterpropagate, the magneto-opticFaraday effect induces a phase difference equal to 2θ between thecounterpropagating waves. If a magnetic field is applied to a fibercoil, however, the clockwise and counterclockwise waves will in generalexperience a slightly different phase shift. The result is amagnetic-field-induced relative phase shift between the clockwise andcounterclockwise propagating waves at the output of the fiber optic loopwhere the waves interfere. This differential phase shift is proportionalto the Verdet constant. This phase difference also depends on themagnitude of the magnetic field and the birefringence of the fiber inthe loop. Additionally, the phase shift depends on the orientation(i.e., the direction) of the magnetic field with respect to the fiberoptic loop as well as on the polarizations of the clockwise andcounterclockwise propagating signals. If the magnetic field is DC, thisdifferential phase shift results in a DC offset in the phase bias of theSagnac interferometer. If the magnetic field varies over time, thisphase bias drifts, which is generally undesirable and thus notpreferred.

The earth's magnetic field poses particular difficulty for Sagnacinterferometers employed in navigation. For example, as an aircrafthaving a fiber optic gyroscope rotates, the relative spatial orientationof the fiber optic loop changes with respect to the magnetic field ofthe earth. As a result, the phase bias of the output of the fibergyroscope drifts. This magnetic field-induced drift can be substantialwhen the fiber optic loop is sufficiently long, e.g., about 1000 meters.To counter the influence of the magnetic field in inertial navigationfiber optic gyroscopes, the fiber optic loop may be shielded fromexternal magnetic fields. Shielding comprising a plurality of layers ofμ-metal may be utilized.

SUMMARY OF THE INVENTION

In certain embodiments, an optical sensor is provided. The opticalsensor comprises a directional coupler comprising at least a first port,a second port, and a third port. The first port is in opticalcommunication with the second port and with the third port such that afirst optical signal received by the first port is split into a secondoptical signal that propagates to the second port and a third opticalsignal that propagates to the third port. The optical sensor furthercomprises a photonic bandgap fiber having a hollow core and an innercladding generally surrounding the core. The photonic bandgap fiber isin optical communication with the second port and with the third port.The second optical signal and the third optical signal counterpropagatethrough the photonic bandgap fiber and return to the third port and thesecond port, respectively. The photonic bandgap fiber has a phasethermal constant S less than 8 parts-per-million per degree Celsius.

In certain embodiments, a method for sensing is provided. The methodcomprises providing a light signal. The method further comprisespropagating a first portion of the light signal in a first directionthrough a portion of a photonic bandgap fiber having a hollow core andan inner cladding generally surrounding the core. The photonic bandgapfiber has a phase thermal constant S less than 8 parts-per-million perdegree Celsius. The method further comprises propagating a secondportion of the light signal in a second direction through the photonicbandgap fiber, the second direction opposite to the first direction. Themethod further comprises optically interfering the first and secondportions of the light signal after the first and second portions of thelight signal propagate through the photonic bandgap fiber, therebyproducing an optical interference signal. The method further comprisessubjecting at least a portion of the photonic bandgap fiber to aperturbation. The method further comprises measuring variations in theoptical interference signal caused by the perturbation.

In certain embodiments, an optical system is provided. The opticalsystem comprises a light source having an output that emits a firstoptical signal. The optical system further comprises a directionalcoupler comprising at least a first port, a second port and a thirdport. The first port is in optically communication with the light sourceto receive the first optical signal emitted from the light source. Thefirst port is in optical communication with the second port and with thethird port such that the first optical signal received by the first portis split into a second optical signal that propagates to the second portand a third optical signal that propagates to the third port. Theoptical system further comprises a photonic bandgap fiber having ahollow core, an inner cladding generally surrounding the core, an outercladding generally surrounding the inner cladding, and a jacketgenerally surrounding the outer cladding. The photonic bandgap fiber isin optical communication with the second port and with the third port.The second optical signal and the third optical signal counterpropagatethrough the photonic bandgap fiber and return to the third port and thesecond port, respectively. The photonic bandgap fiber has a phasethermal constant S less than 8 ppm per degree Celsius. The opticalsystem further comprises an optical detector in optical communicationwith the directional coupler. The optical detector receives thecounterpropagating second optical signal and the third optical signalafter having traversed the photonic bandgap fiber.

BRIEF DESCRIPTION OF THE DRAWINGS

Various embodiments are described below in connection with theaccompanying drawings, in which:

FIG. 1 is a schematic drawing of an example fiber optic sensor depictingthe light source, the fiber loop, and the optical detector.

FIG. 2A is a partial perspective view of the core and a portion of thesurrounding cladding of an example hollow-core photonic-bandgap fiberthat can be used in the example fiber optic sensor.

FIG. 2B is a cross-sectional view of the example hollow-corephotonic-bandgap fiber of FIG. 2A showing more of the features in thecladding arranged in a pattern around the hollow core.

FIG. 3 is a schematic drawing of an example Sagnac interferometerwherein the light source comprises a narrowband light source.

FIG. 4 is a schematic drawing of an example Sagnac interferometer drivenby a narrowband light source with a modulator for modulating theamplitude of the narrowband light source.

FIG. 5 is a schematic drawing of an example Sagnac interferometerwherein the light source comprises a broadband light source.

FIG. 6 is a schematic drawing of an example Sagnac interferometer drivenby a broadband light source with a modulator for modulating theamplitude of the broadband light source.

FIG. 7 schematically illustrates an experimental configuration of anall-fiber air-core PBF gyroscope in accordance with certain embodimentsdescribed herein.

FIG. 8A shows a scanning electron micrograph of a cross-section of anair-core fiber.

FIG. 8B illustrates the measured transmission spectrum of an air-corefiber and a source spectrum from a light source.

FIGS. 9A and 9B show typical oscilloscope traces of these signals, withand without rotation, respectively.

FIG. 10 shows a trace of the f output signal recorded over one hourwhile the gyroscope was at rest.

FIG. 11 schematically illustrates a cross-section of a cylindrical fiberwith a hollow core, a honeycomb inner cladding, an outer cladding, and ajacket.

FIG. 12 illustrates the computed radial deformation as a function ofdistance from fiber center for the Crystal Fibre PBF. The inset is amagnification of the radial deformation over the inner claddinghoneycomb.

FIG. 13 shows the calculated dependence of S, S_(n), and S_(L) on thenormalized core radius for an air-core fiber (solid curves) and for anSMF28 fiber (reference levels) at 1.5 microns.

FIGS. 14A and 14B illustrate SEM photographs of cross-sections of twofibers compatible with certain embodiments described herein.

FIG. 15A schematically illustrates an experimental Michelsoninterferometer used to measure the thermal constant of individualfibers.

FIG. 15B schematically illustrates an experimental Michelsoninterferometer used to measure the error in the fringe count due toresidual heating of the non-PBF portions of the interferometer.

FIGS. 16A and 16B illustrate the measured output power P_(out)(t) andthe measured temperature T(t) for a Blaze Photonics fiber.

FIG. 17 illustrates the value of S_(L) versus jacket thickness for a PBFwith the same crystal period and core radius as the Blaze Photonics PBFfor various air filling ratios.

FIG. 18 illustrates the calculated values of S_(L) for the same fiberstructure as the Blaze Photonics PBF but with different jacket materialsand thicknesses.

FIG. 19 illustrates the calculated dependence of S_(L) on the thicknessof the silica outer cladding.

FIG. 20 schematically illustrates an example configuration for testing afiber optic gyroscope compatible with certain embodiments describedherein.

FIG. 21 shows the measured dependence of random walk on the input signalpower of an air-core fiber gyroscope.

FIG. 22A shows the temperature change applied to the air-core fibergyroscope (dashed line) and the measured resulting change in outputsignal (solid line).

FIG. 22B shows the time derivative of the applied temperature change(dashed line) of FIG. 22A with the measured resulting change in outputsignal of FIG. 22A (solid line).

FIG. 23 schematically illustrates the quadrupolar winding of the firstfew layers of a FOG coil.

FIG. 24A shows the temperature change applied to the conventionalsolid-core fiber gyroscope (dashed line) and the measured resultingchange in output signal (solid line).

FIG. 24B shows the time derivative of the applied temperature change(dashed line) of FIG. 24A with the measured resulting change in outputsignal of FIG. 24A (solid line).

FIG. 25 shows the dependence of the maximum rotation rate error on theapplied temperature gradient measured in both the conventionalsolid-core fiber gyroscope and in the air-core fiber gyroscope.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

A need exists to reduce or eliminate the noise and/or phase driftinduced by Rayleigh backscattering, the Kerr effect, and themagneto-optic Faraday effect present in a fiber interferometer, as wellas other accuracy-limiting effects. In accordance with certainembodiments disclosed herein, a hollow-core photonic-bandgap opticalfiber is incorporated in a fiber optic sensor (e.g., a Sagnacinterferometer) to improve performance or to provide other designalternatives. While certain embodiments described herein utilize aSagnac interferometer, fiber optic sensors utilizing other types ofinterferometers (e.g., Mach-Zehnder interferometers, Michelsoninterferometers, Fabry-Perot interferometers, ring interferometers,fiber Bragg gratings, long-period fiber Bragg gratings, and Fox-Smithinterferometers) can also have improved performance by utilizing ahollow-core photonic-bandgap optical fiber. Fiber optic sensorsutilizing interferometry can be used to detect a variety ofperturbations to at least a portion of the optical fiber. Suchperturbation sensors can be configured to be sensitive to magneticfields, electric fields, pressure, displacements, rotations, twisting,bending, or other mechanical deformations of at least a portion of thefiber.

FIG. 1 illustrates an example Sagnac interferometer 5 that comprises afiber optic system 12 that incorporates a photonic-bandgap fiber 13,which, in certain embodiments, is a hollow-core photonic-bandgap fiber.A version of a similar fiber optic system that includes a conventionaloptical fiber rather than a photonic-bandgap fiber is more fullydescribed in U.S. Pat. No. 4,773,759 to Bergh et al., issued on Sep. 27,1988, which is hereby incorporated herein by reference in its entirety.

The fiber optic system 12 includes various components positioned atvarious locations along the fiber optic system 12 for guiding andprocessing the light. Such components and their use in a Sagnacinterferometer 5 are well-known. Alternative embodiments of the system12 having similar designs or different designs may be realized by thoseskilled in the art and used in certain embodiments described herein.

As configured for the Sagnac rotation sensor 5 in FIG. 1, the fiberoptic system 12 includes a light source 16, a fiber optic loop 14 formedwith the hollow-core photonic-bandgap fiber 13 (described below inconnection with FIGS. 2A and 2B), and a photodetector 30. The wavelengthof the light output from the light source 16 may be approximately 1.50to 1.58 microns, in a spectral region where the loss of silica-basedoptical fibers is near its minimum. Other wavelengths, however, arepossible, and the wavelength of the source emission is not limited tothe wavelengths recited herein. For example, if the optical fibercomprises a material other than silica, the wavelength is advantageouslychosen in the range of wavelengths that minimizes or reduces the losscaused by the optical fiber. Additional detail regarding the lightsource of various embodiments are described in further detail below.

The fiber loop 14 in the optic fiber system 12 in certain embodimentsadvantageously comprises a plurality of turns of the photonic-bandgapfiber 13, which is wrapped in certain embodiments about a spool or othersuitable support (not shown). By way of specific example, the loop 14may comprise more than a thousand turns of the photonic-bandgap fiber 13and may comprise a length of optical fiber 13 of about 1000 meters. Theoptical detector 30 may be one of a variety of photodetectors well knownin the art, although detectors yet to be devised may be used as well.

An optional polarization controller 24 may be advantageously included inthe interferometer as illustrated in FIG. 1. The optional inclusion ofthe polarization controller 24 depends on the design of the system 12.Example polarization controllers are described, for example, in H. C.Lefèvre, Single-Mode Fibre Fractional Wave Devices and PolarisationControllers, Electronics Letters, Vol. 16, No. 20, Sep. 25, 1980, pages778-780, and in U.S. Pat. No. 4,389,090 to Lefèvre, issued on Jun. 21,1983, which are hereby incorporated by reference herein in theirentirety. The polarization controller 24 permits adjustment of the stateof polarization of the applied light. Other types of polarizationcontrollers may be advantageously employed.

The polarization controller 24 shown in FIG. 1 is optically coupled to aport A of a directional coupler 26. The directional coupler 26 coupleslight received by port A to a port B and to a port D of the coupler 26.A port C on the coupler 26 is optically coupled to the photodetector 30.Light returning from the Sagnac interferometer is received by port B andis optically coupled to port A and to port C. In this manner, returninglight received by port B is detected by the photodetector 30 opticallycoupled to port C. As shown, port D terminates non-reflectively at thepoint labeled “NC” (for “not connected”). An example coupler that may beused for the coupler 26 is described in detail in U.S. Pat. No.4,536,058 to Shaw et al., issued on Aug. 20, 1985, and in EuropeanPatent Publication No. 0 038 023, published on Oct. 21, 1981, which areboth incorporated herein by reference in their entirety. Other types ofoptical couplers, however, such as fused couplers, integrated opticalcouplers, and couplers comprising bulk optics may be employed as well.

Port B of the directional coupler 26 is optically coupled to a polarizer32. After passing through the polarizer 32, the optical path of thesystem 12 continues to a port A of a second directional coupler 34. Thecoupler 34 may be of the same type as described above with respect tothe first directional coupler 26 but is not so limited, and may compriseintegrated-optic or bulk-optic devices. In certain embodiments, thelight entering port A of the coupler 34 is divided substantially equallyas it is coupled to a port B and a port D. A first portion W1 of thelight exits from port B of the coupler 34 and propagates around the loop14 in a clockwise direction as illustrated in FIG. 1. A second portionW2 of the light exits from port D of the coupler 34 and propagatesaround the loop 14 in a counterclockwise direction as illustrated inFIG. 1. As shown, port C of the coupler 34 terminates non-reflectivelyat a point labeled “NC.” In certain embodiments, the second coupler 34functions as a beam-splitter to divide the applied light into the twocounterpropagating waves W1 and W2. Further, the second coupler 34 ofcertain embodiments also recombines the counterpropagating waves afterthey have traversed the loop 14. As noted above, other types ofbeam-splitting devices may be used instead of the fiber opticdirectional couplers 26, 34 depicted in FIG. 1.

The coherent backscattering noise in a fiber optic gyroscope using anasymmetrically located phase modulator to provide bias can besubstantially reduced or eliminated by selecting the coupling ratio ofthe coupler 34 to precisely equal to 50%. See, for example, J. M.Mackintosh et al., Analysis and observation of coupling ratio dependenceof Rayleigh backscattering noise in a fiber optic gyroscope, Journal ofLightwave Technology, Vol. 7, No. 9, September 1989, pages 1323-1328.This technique of providing a coupling efficiency of 50% can beadvantageously used in the Sagnac interferometer 5 of FIG. 1 thatutilizes the photonic-bandgap fiber 13 in the loop 14. In aphotonic-bandgap fiber and in a Bragg fiber with a hollow (e.g.,gas-filled) core, backscattering originates from three main sources. Thefirst source is bulk scattering from the gas that fills the hollow core(as well as from the gas that fills the cladding holes), which isnegligible. The second source is bulk scattering from the solid portionsof the waveguide, namely the concentric rings of alternating indexsurrounding the core in a Bragg fiber, and the solid membranessurrounding the holes in a photonic-bandgap fiber. The third source issurface scattering occuring at the surface of the solid portions of thewaveguide, in particular the solid that defines the outer edges of thecore, due to irregularities on this surface, or equivalently randomvariations in the dimensions and shapes of these surfaces. With properdesign of the fiber, surface scattering can be minimized by reducing theamplitude of these variations. Under such conditions, surface scatteringdominates bulk scattering, and surface scattering is much lower that ofa conventional solid-core, index-guiding single-mode fiber. Thebackscattering noise in the Sagnac interferometer 5 of FIG. 1 cantherefore be reduced below the level provided by the inherently lowRayleigh backscattering of the photonic-bandgap fiber 13. The Sagnacinterferometer 5 may be advantageously used as a fiber optic gyroscopefor high-rotation-sensitivity applications that require extremely lowoverall noise.

The above-described technique of employing a coupler 34 with a couplingefficiency of 50% works well as long as the coupling ratio of coupler 34remains precisely at 50%. However, as the fiber environment changes(e.g., the coupler temperature fluctuates) or as the coupler 34 ages,the coupling ratio typically varies by small amounts. Under theseconditions, the condition for nulling the coherent backscatteringdescribed in the previous paragraph may not be continuously satisfied.The use of the photonic-bandgap fiber 13 in the loop 14 instead of aconventional fiber, in conjunction with this coupling technique, relaxesthe tolerance for the coupling ratio to be exactly 50%. Thephotonic-bandgap fiber 13 also reduces the backscattering noise levelarising from a given departure of the coupling ratio from its preferredvalue of 50%.

A polarization controller 36 may advantageously be located between thesecond directional coupler 34 and the loop 14. The polarizationcontroller 36 may be of a type similar to the controller 24 or it mayhave a different design. The polarization controller 36 is utilized toadjust the polarization of the waves counterpropagating through the loop14 so that the optical output signal, formed by superposition of thesewaves, has a polarization that will be efficiently passed, with minimaloptical power loss, by the polarizer 32. Thus, by utilizing bothpolarization controllers, 24, 36, the polarization of the lightpropagating through the fiber 12 may be adjusted for maximum opticalpower. Adjusting the polarization controller 36 in this manner alsoguarantees polarization reciprocity. Use of the combination of thepolarizer 32 and the polarization controllers 24, 36 is disclosed inU.S. Pat. No. 4,773,759, cited above. See also, Chapter 3 of HervéLefèvre, The Fiber-Optic Gyroscope, cited above.

In certain embodiments, a first phase modulator 38 is driven by an ACgenerator 40 to which it is connected by a line 41. The phase modulator38 of certain embodiments is mounted on the optical fiber 13 in theoptical path between the fiber loop 14 and the coupler 34. Asillustrated in FIG. 1, the phase modulator 38 is located asymmetricallyin the loop 14. Thus, the modulation of the clockwise propagating waveW1 is not necessarily in phase with the modulation of thecounterclockwise propagating wave W2 because corresponding portions ofthe clockwise wave W1 and the counterclockwise wave W2 pass through thephase modulator at different times. Indeed, the modulation of the wavesmust be out of phase so that the phase modulator 38 provides a means tointroduce a differential phase shift between the two waves. Thisdifferential phase shift biases the phase of the interferometer suchthat the interferometer exhibits a non-zero first-order sensitivity to ameasurand (e.g., a small rotation rate). More particularly, themodulation of the wave W1 of certain embodiments is about 180° out ofphase with the modulation of the wave W2 so that the first-ordersensitivity is maximum or about maximum. Details regarding thismodulation are discussed in U.S. Pat. No. 4,773,759, cited above.

In various embodiments, the amplitude and frequency of the phase appliedby the loop phase modulator 38 can be selected such that the coherentbackscattering noise is substantially cancelled. See, for example, J. M.Mackintosh et al., Analysis and observation of coupling ratio dependenceof Rayleigh backscattering noise in a fiber optic gyroscope, citedabove. This selection technique can be advantageously used in a fiberoptic gyroscope utilizing a photonic-bandgap fiber loop. Thebackscattering noise can thereby be reduced below the level permitted bythe inherently low Rayleigh backscattering of the photonic-bandgapfiber, which may be useful in applications requiring extremely lowoverall noise. Conversely, this technique for selecting amplitude andfrequency of the phase applied by the loop phase modulator 38 works wellas long as the amplitude and frequency of the applied phase remainsprecisely equal to their respective optimum value. The use of aphotonic-bandgap fiber loop instead of a conventional fiber, inconjunction with this technique, relaxes the tolerance on the stabilityof the amplitude and frequency of the phase applied by the loop phasemodulator 38. This selection technique also reduces the backscatteringnoise level that may occur when the amplitude, the frequency, or boththe amplitude and the frequency of the modulation applied by the loopphase modulator 38 vary from their respective preferred values.

In certain embodiments, a second phase modulator 39 is mounted at thecenter of the loop 14. The second phase modulator 39 is driven by asignal generator (not shown). The second phase modulator 39 mayadvantageously be utilized to reduce the effects of backscattered light,as described, for example, in U.S. Pat. No. 4,773,759, cited above. Thesecond phase modulator 39 may be similar to the first phase modulator 38described above, but the second phase modulator of certain embodimentsoperates at a different frequency than the first phase modulator 38, andthe second phase modulator 39 of certain embodiments is not synchronizedwith the first phase modulator 38.

In various embodiments, the photonic-bandgap fiber 13 within the loop 14and the phase modulators 38 and 39 advantageously comprisepolarization-preserving fiber. In such cases, the polarizer 32 may ormay not be excluded, depending on the required accuracy of the sensor.In one embodiment, the light source 16 comprises a laser diode thatoutputs linearly polarized light, and the polarization of this light ismatched to an eigenmode of the polarization maintaining fiber. In thismanner, the polarization of the light output from the laser diode 10 maybe maintained in the fiber optic system 12.

The output signal from the AC generator 40 is shown in FIG. 1 as beingsupplied on a line 44 to a lock-in amplifier 46, which also is connectedvia a line 48 to receive the electrical output of the photodetector 30.The signal on line 44 to the amplifier 46 provides a reference signal toenable the lock-in amplifier 46 to synchronously detect the detectoroutput signal on line 48 at the modulation frequency of the phasemodulator 38. Thus, the lock-in amplifier 46 of certain embodimentseffectively provides a band-pass filter at the fundamental frequency ofthe phase modulator 38 that blocks all other harmonics of thisfrequency. The power in this fundamental component of the detectedoutput signal is proportional, over an operating range, to the rotationrate of the loop 14. The lock-in amplifier 46 outputs a signal, which isproportional to the power in this fundamental component, and thusprovides a direct indication of the rotation rate, which may be visuallydisplayed on a display panel 47 by supplying the lock-in amplifieroutput signal to the display panel 47 on a line 49. Note that in otherembodiments, the lock-in amplifier may be operated in different modes ormay be excluded altogether, and the signal can be detected byalternative methods. See, for example, B. Y. Kim, Signal ProcessingTechniques, Optical Fiber Rotation Sensing, William Burns, Editor,Academic Press, Inc., 1994, Chapter 3, pages 81-114.

Optical Fibers

As is well known, conventional optical fibers comprise a high indexcentral core surrounded by a lower index cladding. Because of the indexmismatch between the core and cladding, light propagating within a rangeof angles along the optical fiber core is totally internally reflectedat the core-cladding boundary and thus is guided by the fiber core.Typically, although not always, the fiber is designed such that asubstantial portion of the light remains within the core. As describedbelow, the photonic-bandgap fiber 13 in the optical loop 14 also acts asa waveguide; however, the waveguide is formed in a different manner, andits mode properties are such that various effects that limit theperformance of a fiber interferometer that uses conventional fiber(e.g., a Sagnac interferometer) can be reduced by using thephotonic-bandgap fiber 13 in portions of the fiber optic system 12,particularly in the optical loop 14.

An example hollow-core photonic-bandgap fiber 13 is shown in FIGS. 2Aand 2B. Hollow-core photonic-bandgap fibers (photonic crystal fibers)are well-known. See, for example, U.S. Pat. No. 5,802,236 to DiGiovanniet al, issued on Sep. 1, 1998, for Article Comprising a MicrostructureOptical Fiber, and Method of Making such Fiber; U.S. Pat. No. 6,243,522to Allen et al., issued on Jun. 5, 2001, for Photonic Crystal Fibers;U.S. Pat. No. 6,260,388 to Borrelli et al., issued on Jul. 17, 2001, forMethod of Fabricating Photonic Glass Structures by Extruding, Sinteringand Drawing; U.S. Pat. No. 6,334,017 to West et al, issued on Dec. 25,2001, for Ring Photonic Crystal Fibers; and U.S. Pat. No. 6,334,019 toBirks et al., issued on Dec. 25, 2001, for Single Mode Optical Fiber,which are hereby incorporated herein by reference in their entirety.

As illustrated in FIGS. 2A and 2B, the hollow-core photonic-bandgapfiber 13 includes a central core 112. A cladding 114 surrounds the core112. Unlike the central core of conventional fiber, the central core 112of the fiber 13 of certain embodiments is hollow. The open region withinthe hollow core 112 may be evacuated or it may be filled with air orother gases. The cladding 114 includes a plurality of features 116arranged in a periodic pattern so as to create a photonic-bandgapstructure that confines light to propagation within the hollow core 112.For example, in the example fiber 13 of FIGS. 2A and 2B, the features116 are arranged in a plurality of concentric triangles around thehollow core 112. The two innermost layers of holes in the examplepattern are shown in the partial perspective view of FIG. 2A. A completepattern of four concentric layers of holes is illustrated in thecross-sectional view of FIG. 2B. Although the illustrated hole patternis triangular, other arrangements or patterns may advantageously beused. In addition, the diameter of the core 112 and the size, shape, andspacing of the features 116 may vary.

As illustrated by phantom lines in FIG. 2A, the features 116 mayadvantageously comprise a plurality of hollow tubes 116 formed within amatrix material 118. The hollow tubes 116 are mutually parallel andextend along the length of the photonic-bandgap fiber 13 such that thetubes 116 maintain the triangular grid pattern shown in FIG. 2B. Thematrix material 118 that surrounds each of the tubes 116 comprises, forexample, silica, silica-based materials or various other materials wellknown in the art, as well as light-guiding materials yet to be developedor applied to photonic-bandgap technology.

The features (e.g., holes) 116 are specifically arranged to create aphotonic-bandgap. In particular, the distance separating the features116, the symmetry of the grid, and the size of the features 116 areselected to create a photonic bandgap where light within a range offrequencies will not propagate within the cladding 114 if the claddingwas infinite (i.e., in the absence of the core 112). The introduction ofthe core 112, also referred to herein as a “defect,” breaks the symmetryof this original cladding structure and introduces new sets of modes inthe fiber 13. These modes in the fiber 13 have their energy guided bythe core and are likewise referred to as core modes. The array offeatures (e.g., holes) 116 in certain embodiments is specificallydesigned so as to produce a strong concentration of optical energywithin the hollow core 112. In certain embodiments, light propagatessubstantially entirely within the hollow core 112 of the fiber 13 withvery low loss. Exemplary low loss air core photonic band-gap fiber isdescribed in N. Venkataraman et al., Low Loss (13 dB/km) Air CorePhotonic Band-Gap Fibre, Proceedings of the European Conference onOptical Communication, ECOC 2002, Post-deadline Paper No. PD1.1,September 2002.

In various embodiments, the fiber parameters are further selected sothat the fiber is “single mode” (i.e., such that the core 112 supportsonly the fundamental core mode). This single mode includes in fact thetwo eigenpolarizations of the fundamental mode. The fiber 13 thereforesupports two modes corresponding to both eigenpolarizations. In certainembodiments, the fiber parameters are further selected so that the fiberis a single-polarization fiber having a core that supports andpropagates only one of the two eigenpolarizations of the fundamentalcore mode. In certain embodiments, the fiber is a multi-mode fiber.

Other types of photonic-bandgap fibers or photonic-bandgap devices, bothknown and yet to be devised, may be employed in the Sagnac rotationsensors as well as interferometers employed for other purposes. Forexample, one other type of photonic-bandgap fiber that may beadvantageously used is a Bragg fiber.

In accordance with certain embodiments disclosed herein, a Bragg fiberis incorporated in a fiber optic sensor (e.g., a Sagnac interferometer)to improve performance or to provide other design alternatives. Whilecertain embodiments described herein utilize a Sagnac interferometer,fiber optic sensors utilizing other types of interferometers (e.g.,Mach-Zehnder interferometers, Michelson interferometers, Fabry-Perotinterferometers, ring interferometers, fiber Bragg gratings, long-periodfiber Bragg gratings, and Fox-Smith interferometers) can also haveimproved performance by utilizing a Bragg fiber. Fiber optic sensorsutilizing interferometry can be used to detect a variety ofperturbations to at least a portion of the optical fiber. Suchperturbation sensors can be configured to be sensitive to magneticfields, electric fields, pressure, displacements, rotations, twisting,bending, or other mechanical deformations of at least a portion of thefiber.

A Bragg fiber includes a cladding surrounding a core, wherein thecore-cladding boundary comprises a plurality of thin layers of materialswith alternating high and low refractive indices. In variousembodiments, the cladding interface (i.e., the core-cladding boundary)comprises a plurality of concentric annular layers of materialsurrounding the core. The thin layers act as a Bragg reflector andcontains the light in the low-index core. For example, the core ofcertain embodiments is hollow (e.g., containing a gas or combination ofgases, such as air). Bragg fibers are described, for example, in P. Yehet al., Theory of Bragg Fiber, Journal of Optical Society of America,Vol. 68, 1978, pages 1197-1201, U.S. Pat. No. 7,190,875, U.S. Pat. No.6,625,364, and U.S. Pat. No. 6,463,200, each of which is incorporatedherein by reference in its entirety. For a Bragg fiber, the amount ofbackscattering and backreflection at the interface with a conventionalfiber can be different from that for other types of photonic-bandgapfibers. For angled connections, the amount of backreflection from theinterface of a Bragg fiber and a conventional fiber can depend on theangle, wavelength, and spatial orientation in different ways than forother types of photonic-bandgap fibers. Furthermore, as described morefully below, in certain embodiments, a Bragg fiber advantageouslyprovides reduced phase sensitivity to temperature fluctuations.

The accuracy of a fiber optic gyroscope (FOG) is generally limited by asmall number of deleterious effects that arise from undesirableproperties of the loop fiber, namely Rayleigh backscattering, the Kerr,Faraday, and thermal (Shupe) effects. These effects induce short-termnoise and/or long-term drift in the gyroscope output, which limit theability to accurately measure small rotation rates over long periods oftime. The small uncorrected portions of these deleterious effectsconstitute one of the main remaining obstacles to an inertial-navigationFOG.

The use of hollow-core photonic-bandgap fiber instead of conventionaloptical fiber in a Sagnac interferometer may substantially reduce noiseand error introduced by Rayleigh backscattering, the Kerr effect, andthe presence of magnetic fields. In hollow-core photonic-bandgap fiber,the optical mode power is mostly confined to the hollow core, which maycomprise, for example, air, another gas, or vacuum. Rayleighbackscattering as well as Kerr nonlinearity and the Verdet constant aresubstantially less in air, other gases, and vacuum than in silica,silica-based materials, and other solid optical materials. The reductionof these effects coincides with the increased fraction of the opticalmode power contained in the hollow core of the photonic-bandgap fiber.

The Kerr effect and the magneto-optic effect tend to induce a long-termdrift in the bias point of the Sagnac interferometer, which results in adrift of the scale factor correlating the phase shift with the rotationrate applied to the fiber optic gyroscope. In contrast, Rayleighbackscattering tends to introduce mostly short-term noise in themeasured phase, thereby raising the minimum detectable rotation rate.Each of these effects interferes with the extraction of the desiredinformation from the detected optical signal. The incorporation of thehollow-core photonic-bandgap fiber 13 into the interferometer 5 incertain embodiments advantageously diminishes these effects.

A parameter, η_(nl), is defined herein as the fractional amount offundamental mode intensity squared in the solid portions of thephotonic-bandgap fiber. Similarly, a parameter, η, is defined herein asthe fractional amount of fundamental mode power in the solid portions ofthe photonic-bandgap fiber. The phase drift caused by the Kerrnonlinearity is proportional to the parameter η_(nl), and the phasedrift caused by the magneto-optic effect, as well as the noiseintroduced by Rayleigh backscattering, are each proportional to theparameter, η, provided that η is not too small. An analysis of theeffect of η_(nl) is set forth below for the Kerr effect. Similaranalyses can be performed for Rayleigh backscattering and themagneto-optic Faraday effects, using the parameter η.

Kerr Effect

Since some of the mode energy resides in the holes including the core ofthe photonic-bandgap fiber and some of mode energy resides in the solidportions of the fiber (typically a silica-based glass), the Kerr effectin a photonic-bandgap fiber (PBF) includes two contributions. Onecontribution is from the solid portions of the fiber, and onecontribution is from the holes. The residual Kerr constant of aphotonic-bandgap fiber, n_(2,PBF), can be expressed as the sum of thesetwo contributions according to the following equation:n _(2,PBF) =n _(2,solid)η_(nl) +n _(2,holes)(1−η_(nl))  (1)where n_(2,solid) is the Kerr constant for the solid portion of thefiber, which may comprise for example silica, and where n_(2,holes) isthe Kerr constant for the holes, which may be, for example, evacuated,gas-filled, or air-filled. If the holes are evacuated, the Kerrnonlinearity is zero because the Kerr constant of vacuum is zero. Withthe Kerr constant equal to zero, the second contribution correspondingto the term n_(2,holes)(1−η_(nl)) in Equation (1) is absent. In thiscase, the Kerr nonlinearity is proportional to the parameter, η_(nl), asindicated by the remaining term n_(2,solid)η_(nl). However, if the holesare filled with air, which has small but finite Kerr constant, bothterms (n_(2,solid)η_(nl)+n_(2,holes)(1−η_(nl)) are present. Equation (1)above accounts for this more general case.

In certain embodiments, the parameter η_(nl) is equal toA_(eff)/A_(eff, silica), where A_(eff) and A_(eff, silica) are the modeeffective area and mode effective area in silica, respectively. Thesequantities can be computed as follows:

${A_{eff} = \frac{1}{n_{g}^{2}}}\frac{\left( {\underset{{Waveguidecross}\text{-}{section}}{\int\int}{ɛ_{r}\left( {x,y} \right)}{E\left( {x,y} \right)}^{2}{\mathbb{d}x}{\mathbb{d}y}} \right)^{2}}{\underset{{Waveguidecross}\text{-}{section}}{\int\int}{E\left( {x,y} \right)}^{4}{\mathbb{d}x}{\mathbb{d}y}}$${A_{{eff},{silica}} = \frac{1}{n_{g}^{2}}}\frac{\left( {\underset{{Waveguidecross}\text{-}{section}}{\int\int}{ɛ_{r}\left( {x,y} \right)}{E\left( {x,y} \right)}^{2}{\mathbb{d}x}{\mathbb{d}y}} \right)^{2}}{\underset{silica}{\int\int}{E\left( {x,y} \right)}^{4}{\mathbb{d}x}{\mathbb{d}y}}$where n_(g) is the mode group velocity, ε_(r) is the relativepermittivity, and E is the electric field of the mode. Note that theparameter η_(nl) is the fractional amount of fundamental mode intensitysquared in the solid portions of the photonic-bandgap fiber, not thefractional amount of fundamental mode power in the solid portions of thephotonic-bandgap fiber, which is the regular definition of η and whichis valid for the other properties of the photonic-bandgap fiber.

For standard silica fiber, the percentage of the optical mode containedin the cladding is generally in the range of 10% to 20%. In thehollow-core photonic-bandgap fiber 13, the percentage of the opticalmode in the cladding 114 is estimated to be about 1% or substantiallyless. Accordingly, in the photonic-bandgap fiber 13, the effectivenonlinearity due to the solid portions of the fiber may be decreased bya factor of approximately 20. According to this estimate, by using thehollow-core photonic-bandgap fiber 13, the Kerr effect can be reduced byat least one order of magnitude, and can be reduced much more withsuitable design. Indeed, measurements indicate that the photonic-bandgapfibers can be designed with a parameter η_(nl) small enough that theKerr constant of the solid portion of the fiber, n_(2,solid), isnegligible compared to the hole contribution, n_(2,holes)(1−η_(nl)).Even in the case where n_(2,solid) is much larger than n_(2,holes), thefiber can be designed in such a way that η_(nl) is sufficiently smallthat n_(2,holes)(1−η_(nl)) is larger than n_(2,solid)η_(nl). See, forexample, D. G. Ouzounov et al., Dispersion and nonlinear propagation inair-core photonic-bandgap fibers, Proceedings of the Conf. on Lasers andElectro-optics, Paper CThV5, June 2003.

Backscattering and Magneto-Optic Effects

A relationship similar to Equation (1) applies to Rayleighbackscattering and magneto-optic Faraday effect. Accordingly, Equation(1) can be written in the following more general form to encompassRayleigh backscattering and the magneto-optic Faraday effect as well asthe Kerr effect:F _(PBF) =F _(solid) η+F _(holes)(1−η)  (2)In Equation (2), F corresponds to any of the respective coefficients,the Kerr constant n₂, the Verdet constant V, or the Rayleigh scatteringcoefficient α_(s). The terms F_(PBF), F_(solid), and F_(holes) representthe appropriate constant for the photonic-bandgap fiber, for the solidmaterial, and for the holes, respectively. For example, when the Kerrconstant n₂ is substituted for F, Equation (2) becomes Equation (1).When the Verdet constant V is substituted for F, Equation (2) describesthe effective Verdet constant of a photonic-bandgap fiber.

The first term of Equation (2), F_(solid)η, arises from the contributionof the solid portion of the fiber, and the second term F_(holes)(1−η)arises from the contribution of the holes. In a conventional fiber, onlythe first term is present. In a photonic-bandgap fiber, both the termfor the solid portion, F_(solid)η, and the term for the hollow portion,F_(holes)(1−η), generally contribute. The contributions of these termsdepend on the relative percentage of mode power in the solid, which isquantified by the parameter η. As discussed above, if η is madesufficiently small through appropriate fiber design, for example, thefirst term F_(solid)η can be reduced to a negligible value and thesecond term F_(holes)(1−η) dominates. This is beneficial becauseF_(holes) is much smaller than F_(solid), which means that the secondterm is small and thus F is small. This second term F_(holes)(1−η) canbe further reduced by replacing the air in the holes with a gas having areduced Kerr constant n₂, a reduced Verdet constant V, a reducedRayleigh scattering coefficient α_(s), or reduced values of all or someof these coefficients. This second term F_(holes)(1−η) can be reduced tozero if the holes in the fiber are evacuated.

As discussed above, the solid contributions to the Rayleighbackscattering, the Kerr-induced phase error, and themagnetic-field-induced phase shift on the optical signal can bedecreased by reducing the parameter η and η_(nl). Accordingly, thephotonic-bandgap fiber is designed so as to reduce these parameters inorder to diminish the solid contributions of Rayleigh backscattering,Kerr nonlinearity, and the magnetic field effects proportionally. Forexample, in particular designs of the hollow-core photonic-bandgapfiber, the value of η may be about 0.003 or lower, although this rangeshould not be construed as limiting. In addition to this bulk scatteringcontribution, surface scattering can provide a larger contribution.

As described above, Rayleigh backscattering in an optical fiber createsa reflected wave that propagates through the fiber in the directionopposite the original direction of propagation of the primary wave thatproduces the backscattering. Since such backscattered light is coherentwith the light comprising the counterpropagating waves W1, W2, thebackscattered light interferes with the primary waves and thereby addsintensity noise to the signal measured by the detector 30.

Backscattering is reduced in certain embodiments by employing thehollow-core photonic-bandgap fiber 13 in the loop 14. As describedabove, the mode energy of the optical mode supported by the hollow-corephotonic-bandgap fiber 13 is substantially confined to the hollow core112. In comparison to conventional solid-core optical fibers, lessscattering results for light propagating through vacuum, air, or gas inthe hollow core 112.

By increasing the relative amount of mode energy in the holes (includingthe hollow core) and reducing the amount of mode energy in the solidportion of the fiber, backscattering in certain embodiments is reduced.Accordingly, by employing the photonic-bandgap fiber 13 in the loop 14of the fiber optic system 12, backscattering can be substantiallyreduced.

A hollow-core fiber in certain embodiments also reduces the effect of amagnetic field on the performance of the interferometer. As discussedabove, the Verdet constant is smaller in air, gases, and vacuum than insolid optical materials such as silica-based glasses. Since a largeportion of the light in a hollow-core photonic-bandgap fiber propagatesin the hollow core, the magneto-optic-induced phase error is reduced.Thus, less magnetic-field shielding is needed.

Light Sources

Laser light comprising a number of oscillatory modes, or frequencies,e.g., light from a superfluorescent fiber source (SFS), may also be usedin the rotation sensing device described herein to provide a lowerrotation rate error than is possible with light from a single-frequencysource under similar conditions. Multimode lasers may also be employedin some embodiments. In particular, the Kerr-induced rotation rate erroris inversely proportional to the number of oscillating modes in thelaser because multiple frequency components cause the self-phasemodulation and cross-phase modulation terms in the Kerr effect to atleast partially average out, thereby reducing the net Kerr-induced phaseerror. A mathematical analysis of this phenomena and examples ofreductions in the Kerr-induced phase error are disclosed in U.S. Pat.No. 4,773,759, cited above.

Although a superfluorescent light source may be used with the fiberoptic system 12 of FIG. 1, the system 12 of certain embodimentsincorporates a light source 16 that outputs light having a substantiallyfixed single frequency. Because the scale factor of a fiber opticgyroscope depends on the source mean wavelength, random variations inthis wavelength will lead to random variations in the wavelength factor,which introduces undesirable error in the measured rotation rate. Lightsources having a substantially stable output wavelength have beendeveloped for telecommunications applications, and these sources arethus available for use in fiber optic rotation sensing systems. Theselight sources, however, are typically narrowband sources. Accordingly,utilization of these narrowband stable-frequency light sources with aconventional optical fiber would be inconsistent with theabove-described use of broadband multimode laser sources to compensatefor the Kerr effect.

However, FIG. 3 illustrates an interferometer 305 in accordance withcertain embodiments described herein that can achieve a substantiallystable wavelength while reducing the Kerr contributions to the drift inthe interferometer bias. The interferometer 305 comprises an opticalfiber system 312 that includes a stable-frequency narrowband lightsource 316 in combination with the hollow-core photonic-bandgap fiber13. By introducing the hollow-core photonic-bandgap fiber 13 into thefiber optic system 312, the conventionally available narrowband lightsource 316 having a substantially stable-frequency output can beadvantageously used. The Sagnac interferometer 305 in FIG. 3 is similarto the Sagnac interferometer 5 of FIG. 1, and like elements from FIG. 1are identified with like numbers in FIG. 3. As described above withrespect to the fiber optic system 12 of FIG. 1, the fiber optic system312 of FIG. 3 also includes an optical loop 14 that comprises a lengthof the hollow-core photonic bandgap fiber 13. The narrowband lightsource 316 advantageously comprises a light-emitting device 310 such asa laser or other coherent light source. Examples of a light-emittinglaser 310 include a laser diode, a fiber laser, or a solid-state laser.In certain embodiments, operating the FOG with a narrow-band laser diodeadvantageously offers significant advantages over the current broadbandsources, including a far greater wavelength stability and thusscale-factor stability, and possibly a lower cost. Other lasers or othertypes of narrowband light sources may also be advantageously employed inother embodiments. In some embodiments, the narrowband light source 316outputs light having an example FWHM spectral bandwidth of about 1 GHzor less, of about 100 MHz or less, or about 10 MHz or less. Lightsources having bandwidths outside the cited ranges may also be includedin other embodiments.

As discussed above, the light source 316 of certain embodiments operatesat a stable wavelength. The output wavelength may, for example, notdeviate more than about ±10⁻⁶ (i.e., ±1 part per million (ppm)) in someembodiments. The wavelength instability is about ±10⁻⁷ (i.e., ±0.1 ppm)or lower in certain embodiments. Narrowband light sources that offersuch wavelength stability, such as the lasers produced widely fortelecommunication applications, are currently available. Accordingly, asa result of the use of a stable-wavelength light source, the stabilityof the Sagnac interferometer scale factor is enhanced.

A narrowband light source will also result in a longer coherence lengthin comparison with a broadband light source and will thus increase thecontribution of noise produced by coherent backscattering. For example,if the clockwise propagating light signal W1 encounters a defect in theloop 14, the defect may cause light from the light signal W1 tobackscatter in the counterclockwise direction. The backscattered lightwill combine and interfere with light in the counterclockwisepropagating primary light signal W2. Interference will occur between thebackscattered W1 light and the counterclockwise primary light W2 if theoptical path difference traveled by these two light signals isapproximately within one coherence length of the light. For scatterpoints farther away from the center of the loop 14, this optical pathdifference will be largest. A larger coherence length therefore causesscatter points farther and farther away from the center of the loop 14to contribute to coherent noise in the optical signal, which increasesthe noise level.

In certain embodiments, a coherence length which is less than the lengthof the optical path from port B of the coupler 34 to port D would reducethe magnitude of the coherent backscatter noise. However, a narrowbandlight source, such as the narrowband source 316, has a considerablylonger coherence length than a broadband light source and thus willcause more coherent backscatter if a conventional optical fiber is usedinstead of the photonic-bandgap fiber 13 in the embodiment of FIG. 3.However, by combining the use of the stable-frequency narrowband lightsource 316 with the hollow-core photonic-bandgap fiber 13 as shown inFIG. 3, the coherent backscattering can be decreased because thehollow-core photonic-bandgap fiber 13 reduces scattering as describedabove. The bandwidth of the narrowband source 316 of certain embodimentsis selected such that the optical power circulating in either directionthrough the optical loop 14 is smaller than the threshold power forstimulated Brillouin scattering calculated for the specific fiber usedin the coil.

By employing the narrowband stable wavelength optical source 316 inconjunction with the hollow-core photonic-bandgap fiber 13 in accordancewith FIG. 3, scale factor instability resulting from the fluctuatingsource mean wavelength can be decreased while reducing the contributionsof the Kerr nonlinearities as well as coherent backscattering.

If the Kerr effect is still too large and thus introduces a detrimentalphase drift that degrades the performance of the fiber optic system 312of FIG. 3, other methods can also be employed to reduce the Kerr effect.One such method is implemented in a Sagnac interferometer 405illustrated in FIG. 4. The Sagnac interferometer 405 includes a fiberoptic system 412 and a narrowband source 416. The narrowband source 416of FIG. 4 comprises a light-emitting device 410 in combination with anamplitude modulator 411. The light-emitting device 410 mayadvantageously be similar to or the same as the light-emitting device310 of FIG. 3. The optical signal from the light-emitting device 310 ismodulated by the amplitude modulator 411. In certain embodiments, theamplitude modulator 411 produces a square-wave modulation, and, incertain embodiments, the resulting light output from the narrowbandsource 416 has a modulation duty cycle of about 50%. The modulation ismaintained in certain embodiments at a sufficiently stable duty cycle.As discussed, for example, in U.S. Pat. No. 4,773,759, cited above, andin R. A. Bergh et al, Compensation of the Optical Kerr Effect inFiber-Optic Gyroscopes, Optics Letters, Vol. 7, 1992, pages 282-284,such square-wave modulation effectively cancels the Kerr effect in afiber-optic gyroscope. Alternatively, as discussed, for example, inHervé Lefèvre, The Fiber-Optic Gyroscope, cited above, other modulationsthat produce a modulated signal with a mean power equal to the standarddeviation of the power can also be used to cancel the Kerr effect. Forexample, the intensity of the light output from the light source 416 maybe modulated by modulating the electrical current supplied to thelight-emitting device 410.

In certain embodiments, other techniques can be employed in conjunctionwith the use of a narrowband light source 416 of FIG. 4, for example, toreduce noise and bias drift. For example, frequency components can beadded to the narrowband light source 416 by frequency or phasemodulation to effectively increase the bandwidth to an extent. If, forexample, the narrowband light source 416 has a linewidth of about 100MHz, a 10-GHz frequency modulation will increase the laser linewidthapproximately 100 times, to about 10 GHz. Although a 10-GHz modulationis described in this example, the frequency modulation does not need tobe limited to 10 GHz, and may be higher or lower in differentembodiments. The phase noise due to Rayleigh backscattering is inverselyproportional to the square root of the laser linewidth. Accordingly, anincrease in linewidth of approximately 100 fold results in a 10-foldreduction in the short-term noise induced by Rayleigh backscattering.Refinements in the design of the photonic-bandgap fiber 13 to furtherreduce the parameter η can also be used to reduce the noise due toRayleigh scattering to acceptable levels.

FIG. 5 illustrates an embodiment of a Sagnac interferometer 505 thatincorporates a broadband source 516 that may be advantageously used inconjunction with the hollow-core photonic-bandgap fiber 13 in an opticalfiber system 512 in order to mitigate Kerr non-linearity, Rayleighbackscattering and magnetic-field effects. Accordingly, the bias driftas well as the short-term noise can be reduced in comparison to systemsutilizing narrowband light sources.

The broadband light source 516 advantageously comprises a broadbandlight-emitting device 508 such as, for example, a broadband fiber laseror a fluorescent light source. Fluorescent light sources includelight-emitting diodes (LEDs), which are semiconductor-based sources, andsuperfluorescent fiber sources (SFS), which typically utilize arare-earth-doped fiber as the gain medium. An example of a broadbandfiber laser can be found in K. Liu et al., Broadband Diode-Pumped FiberLaser, Electron. Letters, Vol. 24, No. 14, July 1988, pages 838-840.Erbium-doped superfluorescent fiber sources can be suitably employed asthe broadband light-emitting device 508. Several configurations ofsuperfluorescent fiber sources are described, for example, in Rare EarthDoped Fiber Lasers and Amplifiers, Second Edition, M. J. F. Digonnet,Editor, Marcel Dekker, Inc., New York, 2001, Chapter 6, and referencescited therein. This same reference and other references well-known inthe art disclose various techniques that have been developed to produceEr-doped superfluorescent fiber sources with highly stable meanwavelengths. Such techniques are advantageously used in variousembodiments to stabilize the scale factor of the Sagnac interferometer505. Other broadband light sources 516 may also be used.

In certain embodiments, the broadband light source 516 outputs lighthaving a FWHM spectral bandwidth of, for example, at least about 1nanometer. In other embodiments, the broadband light source 516 outputslight having a FWHM spectral bandwidth of, for example, at least about10 nanometers. In certain embodiments, the spectral bandwidth may bemore than 30 nanometers. Light sources having bandwidths outside thedescribed ranges may be included in other embodiments.

In certain embodiments, the bandwidth of the broadband light source canbe reduced to relax design constraints in producing the broadbandsource. Use of the hollow-core photonic-bandgap fiber 13 in the Sagnacinterferometer 505 may at least partially compensate for the increasederror resulting from reducing the number of spectral components thatwould otherwise be needed to help average out the backscatter noise andother detrimental effects. The Sagnac interferometer 505 has less noiseas a result of Kerr compensation and reduced coherent backscattering. Incertain embodiments, the fiber optic system 512 operates with enhancedwavelength stability. The system 512 also possesses greater immunity tothe effect of magnetic fields and may therefore employ less magneticshielding.

The fiber optic system 512 of FIG. 5 advantageously counteracts phaseerror and phase drift, and it provides a high level of noise reduction.This enhanced accuracy may exceed requirements for current navigationaland non-navigational applications.

FIG. 6 illustrates an example Sagnac interferometer 605 in accordancewith certain embodiments described herein. The Sagnac interferometer 605comprises an optical fiber system 612 in combination with a broadbandlight source 616. The broadband source 616 advantageously comprises abroadband light-emitting device 608 in combination with a modulator 611.In certain embodiments, the modulator 611 modulates the power of thebroadband light at a duty cycle of approximately 50%. The modulatedbroadband light from the broadband source 616 contributes to thereduction or elimination of the Kerr effect, as discussed above.

Other advantages to employing a hollow-core photonic-bandgap fiber arepossible. For example, reduced sensitivity to radiation hardening may bea benefit. Silica fiber will darken when exposed to high-energyradiation, such as natural background radiation from space or theelectromagnetic pulse from a nuclear explosion. Consequently, the signalwill be attenuated. In a hollow-core photonic-bandgap fiber, a smallerfraction of the mode energy propagates in silica and thereforeattenuation resulting from exposure to high-energy radiation is reduced.

The example Sagnac interferometers 5, 305, 405, 505 and 605 illustratedin FIGS. 1, 3, 4, 5 and 6 have been used herein to describe theimplementation and benefits of the hollow-core bandgap optical fiber 13of FIGS. 2A and 2B to improve the performances of the interferometers.It should be understood that the disclosed implementations are examplesonly. For example, the interferometers 5, 305, 405, 505 and 605 need notcomprise a fiber optic gyroscope or other rotation-sensing device. Thestructures and techniques disclosed herein are applicable to other typesof sensors or other systems using fiber Sagnac interferometers as well.

Although gyroscopes for use in inertial navigation, have been discussedabove, hollow-core photonic-bandgap fiber can be employed in othersystems, sub-systems, and sensors using a Sagnac loop. For example,hollow-core photonic-bandgap fiber may be advantageously used in fiberSagnac perimeter sensors that detect movement of the fiber and intrusionfor property protection and in acoustic sensor arrays sensitive topressure variations applied to the fiber. Perimeter sensors aredescribed, for example, in M. Szustakowski et al., Recent development offiber optic sensors for perimeter security, Proceedings of the 35thAnnual 2001 International Carnahan Conference on Security Technology,16-19 Oct. 2001, London, UK, pages 142-148, and references citedtherein. Sagnac fiber sensor arrays are described in G. S. Kino et al, APolarization-based Folded Sagnac Fiber-optic Array for Acoustic Waves,SPIE Proceedings on Fiber Optic Sensor Technology and Applications 2001,Vol. 4578 (SPIE, Washington, 2002), pages 336-345, and references citedtherein. Various applications described herein, however, relate to fiberoptic gyroscopes, which may be useful for navigation, to provide a rangeof accuracies from low accuracy such as for missile guidance to highaccuracy such as aircraft navigation. Nevertheless, other uses, boththose well-known as well as those yet to be devised, may also benefitfrom the advantages of various embodiments described herein. Thespecific applications and uses are not limited to those recited herein.

Also, other designs and configurations, those both well known in the artand those yet to be devised, may be employed in connection with theinnovative structures and methods described herein. The interferometers5, 305, 405, 505 and 605 may advantageously include the same ordifferent components as described above, for example, in connection withFIGS. 1, 3, 4, 5 and 6. A few examples of such components includepolarizers, polarization controllers, splitters, couplers, phasemodulators, and lock-in amplifiers. Other devices and structures may beincluded as well.

In addition, the different portions of the optical fiber systems 12,312, 412, 512 and 612 may comprise other types of waveguide structuressuch as integrated optical structures comprising channel or planarwaveguides. These integrated optical structures may, for example,include integrated-optic devices optically connected via segments ofoptical fiber. Portions of the optical fiber systems 12, 312, 412, 512and 612 may also include unguided pathways through free space. Forexample, the optical fiber systems 12, 312, 412, 512 and 612 may includeother types of optical devices such as bulk-optic devices havingpathways in free space where the light is not guided as in a waveguideas well as integrated optical structures. However, much of the opticalfiber system of certain embodiments includes optical fiber whichprovides a substantially continuous optical pathway for light to travelbetween the source and the detector. For example, photonic-bandgap fibermay advantageously be used in portions of the optical fiber systems 12,312, 412, 512 and 612 in addition to the fiber 13 in the loop 14. Incertain embodiments, the entire optical fiber system from the source toand through the loop and back to the detector may comprisephotonic-bandgap fiber. Some or all of the devices described herein mayalso be fabricated in hollow-core photonic-bandgap fibers, followingprocedures yet to be devised. Alternatively, photonic-bandgap waveguidesand photonic-bandgap waveguide devices other than photonic-bandgap fibermay be employed for certain devices.

Several techniques have been described above for lowering the level ofshort-term noise and bias drift arising from coherent backscattering,the Kerr effect, and magneto-optic Faraday effect. It is to beunderstood that these techniques can be used alone or in combinationwith each other in accordance with various embodiments described herein.Other techniques not described herein may also be employed in operatingthe interferometers and to improve performance. Many of these techniquesare well known in the art; however, those yet to be developed areconsidered possible as well. Also, reliance on any particular scientifictheory to predict a particular result is not required. In addition, itshould be understood that the methods and structures described hereinmay improve the Sagnac interferometers in other ways or may be employedfor other reasons altogether.

Temperature (Shupe) Effects

The optical phase of a signal traveling in a conventional optical fiberis a relatively strong function of temperature. As the temperaturechanges, the fiber length, radius, and refractive indices all change,which results in a change of the signal phase. This effect is generallysizable and detrimental in phase-sensitive fiber systems such as thefiber sensors utilizing conventional fibers. For example, in a fibersensor based on a Mach-Zehnder interferometer with 1-meter long arms, atemperature change in one of the arms as small as 0.01 degrees Celsiusis sufficient to induce a differential phase change between the two armsas large as about 1 radian. This is about a million times larger thanthe typical minimum detectable phase of an interferometric sensor (about1 microradian). Dealing with this large phase drift is often asignificant challenge.

A particularly important fiber optic sensor where thermal effects havebeen troublesome is the fiber optic gyroscope (FOG). Although the FOGutilizes an inherently reciprocal Sagnac interferometer, even a smallasymmetric change in the temperature distribution of the Sagnac coilfiber will result in a differential phase change between the twocounter-propagating signals, a deleterious effect known as the Shupeeffect. See, e.g., D. M. Shupe, “Thermally induced nonreciprocity in thefiber-optic interferometer,” Appl. Opt., Vol. 19, No. 5, pages 654-655(1980); D. M. Shupe, “Fibre resonator gyroscope: sensitivity and thermalnonreciprocity,” Appl. Opt., Vol. 20, No. 2, pages 286-289 (1981).Because the Sagnac is a common-path interferometer, the two signals seealmost the same thermally induced change and this differential phasechange is much smaller than in a Mach-Zehnder or Michelsoninterferometer, but it is not small enough for high-accuracyapplications which advantageously have extreme phase stability. In thisand other fiber sensors and systems, thermal effects have beensuccessfully fought with clever engineering solutions. These solutions,however, generally increase the complexity and cost of the finalproduct, and they can also negatively impact its reliability andlifetime.

In certain embodiments, a hollow-core fiber-optic gyroscope has similarshort-term noise as a conventional gyroscope (random walk of about 0.015deg/√hr) and a dramatically reduced sensitivity to Kerr effect (by morethan a factor of 50), temperature transients (by about a factor of 6.5),and Faraday effect (by about a factor greater than 10).

The fundamental mode of a hollow-core photonic-bandgap fiber (PBF)travels mostly in the core containing one or more gases (e.g., air),unlike in a conventional fiber where the mode travels entirely throughsilica. Since gases or combinations of gases (e.g., air) have much lowerKerr nonlinearity and refractive index dependences on temperature thandoes silica of a conventional solid-core fiber, in a hollow-core PBFthese generally deleterious effects are significantly reduced ascompared to a conventional solid-core fiber. See, e.g., D. G. Ouzounov,C. J. Hensley, A. L. Gaeta, N. Venkataraman, M. T. Gallagher and K. W.Koch, “Nonlinear properties of hollow-core photonic band-gap fibers,” inConference on Lasers and Electro-Optics, Optical Society of America,Washington, D.C., Vol. 1, pages 217-219 (2005). Since the thermalcoefficient of the refractive index dn/dT is much smaller for gases thanfor silica, in a hollow-core fiber the temperature sensitivity of themode effective index is reduced considerably. The length of a PBF ofcourse still varies with temperature, which means that the phasesensitivity will not be reduced simply in proportion to the percentageof mode energy in silica. However, it should still be reducedsignificantly, an improvement beneficial to numerous applications,especially in the FOG where it implies a reduced Shupe effect.

This feature can be extremely advantageous in fiber sensors such as thefiber optic gyroscope (FOG), where the Kerr and thermal (Shupe) effectsare notoriously detrimental. By replacing the conventional fiber used inthe sensing coil by a hollow-core fiber, the phase drift induced in aFOG by these two effects should be considerably smaller. Numericalsimulations described herein predict a reduction of about 100-500 foldfor the Kerr effect, and up to about 23 fold for the thermal effect. Forthe very same reason, the gyro's dependence on external magnetic fields(Faraday effect) should be greatly reduced, by a predicted factor ofabout 100-500, as described herein. Relaxing the magnitude of thesethree undesirable effects should result in practice in a significantreduction in the complexity, cost, and yield of commercial FOGs.

Furthermore, if through design and fabrication improvements Rayleighback-scattering in PBFs can be reduced to below that of conventionalsolid-core fibers, it will also be possible to operate a hollow-corefiber gyroscope with a narrow-band communication-grade semiconductorsource instead of the current broadband source (typically an Er-dopedsuperfluorescent fiber source (SFS)). Since it is difficult to stabilizethe mean wavelength of an SFS to better than 1 part-per-million, thischange would offer the additional benefit of improving the source's meanwavelength stability by one to two orders of magnitude, and possiblyreducing the source's cost.

These benefits come at the price of an increased fiber propagation loss(e.g., about 20 dB/km). However, this loss is manageable in practice. Itamounts to only about 4 dB for a 200-meter long coil, which is notexcessive compared to the loss of the other gyroscope components (e.g.,about 15 dB). Furthermore, the state-of-the-art loss of PBFs is likelyto decrease in the future.

EXAMPLE 1

This example describes the operation of an air-core photonic-bandgapfiber gyroscope in accordance with certain embodiments described herein.Because the optical mode in the sensing coil travels largely through airin an air-core photonic-bandgap fiber, which has much smaller Kerr,Faraday, and thermal constants than silica, the air-corephotonic-bandgap fiber has far lower dependencies on power, magneticfield, and temperature fluctuations. With a 235-meter-long fiber coil, aminimum detectable rotation rate of about 2.7°/hour and a long-termstability of about 2°/hour were observed, consistent with the Rayleighbackscattering coefficient of the fiber and comparable to what ismeasured with a conventional fiber. Furthermore, the Kerr effect, theFaraday effect, and Rayleigh backcattering can be reduced by a factor ofabout 100-500, and the Shupe effect by a factor of about 3-11, dependingon the fiber design.

We confirm some of these predictions with the demonstration of the firstair-core fiber gyroscope. In spite of the significantly higher loss andscattering of existing air-core fibers compared to conventional fibers,the sensing performance of this example is comparable to that of aconventional FOG of similar sensing-coil length. This resultdemonstrates that existing air-core fibers can readily improve thegyroscope performance in a number of ways, for example by reducingresidual thermal drifts and relaxing the tolerance on certain componentsand their stabilization.

Using an air-core fiber in the example fiber gyroscope provides areduction of the four deleterious effects discussed above. Referring toEquation (1) which expresses the effective Kerr constant seen by thefundamental core mode, the Kerr constant of air (n_(2,air)≈2.9×10⁻¹⁹cm²/W) is about three orders of magnitude smaller than that of silica(n_(2,silica)≈3.2×10⁻¹⁶ cm²/W). Since η_(nl)<<1, the effective Kerrnonlinearity will be much smaller in an air-core fiber than in aconventional fiber. In fact, the third-order dispersion in a particularair-core fiber n_(2,PBF) is about 250 times smaller than n_(2,silica).

For the Faraday effect, the effective Verdet constant of the fundamentalmode V_(PBF) can be expressed using Equation (1), but with the constantsn₂ replaced by the Verdet constants of silica (V_(silica)) and air(V_(air)). Since V_(air) is much weaker (about 1600 times) thanV_(silica), V_(PBF) should be also reduced by two to three orders ofmagnitudes compared to a conventional fiber.

These considerations show that in a PBF gyro, the bias drifts caused bythe Kerr and Faraday effects are reduced compared to a standardgyroscope roughly in proportion to η, the fractional mode power in thesilica portions of the PBF. The value of η calculated for a single-modeair-core silica PBF ranges approximately from about 0.015 to about0.002, depending on the core radius, air filling ratio, and signalwavelength. Consequently, in a PBF gyroscope both the Kerr-induced andthe Faraday-induced phase drifts can be reduced by a factor of about70-500. These are substantial improvements that should greatly relaxsome of the FOG engineering requirements, such as temperature control ofthe loop coupler and the amount of μ-metal shielding.

An analogous reasoning applies to Rayleigh scattering. Instate-of-the-art silica fibers, the minimum loss is limited by Rayleighscattering and multi-phonon coupling. In contrast, in current air-corefibers loss is believed to be limited by coupling to surface andradiation modes due to random dimensional fluctuations. Thesefluctuations have either a technological origin (such as misalignment ofthe capillary tubes or periodic core diameter variations), or from theformation of surface capillary waves on the silica membranes of the PBFduring drawing due to surface tension. Whereas the former type offluctuations can be reduced with improved manufacturing techniques, thelatter has a more fundamental nature and might be more difficult toreduce, although several approaches are possible. By reducing thesefluctuations to sufficiently low levels, the Rayleigh scatteringcoefficient of air-core PBFs can reach the lower limit imposed bysilica: α_(PBF)=ηα_(silica), where α_(silica) is the scatteringcoefficient of silica and scattering in the air portions of thewaveguide has been neglected. Thus, the lowest possible effectivescattering coefficient for an air-core fiber should be smaller than fora conventional fiber in proportion to η, i.e., again by a factor of70-500. As discussed below, this reduction has a large positive impacton the fiber optic gyroscope.

In a gyroscope, the phase error due to coherence interference betweenthe backscattered waves and the primary waves is given by Equation (3):

$\begin{matrix}{{\delta\phi} = \sqrt{\frac{\Omega}{2\pi}{\eta\alpha}_{silica}L_{c}}} & (3)\end{matrix}$where Ω is the solid angle of the fundamental mode inside the fiber, andL_(c) is the coherence length of the light. In a conventional gyroscopeusing a standard polarization-maintaining (PM) fiber, the externalnumerical aperture (NA) is typically around 0.11, so the internal solidangle is Ω=π(NA/n)²≈0.018, where n is the core refractive index. Themode travels entirely through silica, so η=1, and α_(silica) around 1.5microns is about −105 dB/millimeter, or about 3.2×10⁻⁸ meter⁻¹. Usingthese values in Equation (3) with light supplied by a broadband Er-dopedsuperfluorescent fiber source (SFS) with a coherence length of L_(c)≈230microns yields δφ≈0.15 microradians, illustrating that the backcatteringnoise is small compared to the typical noise of a fiber interferometer.

The backscattering coefficient of a Crystal Fibre's AIR-10-1550 air-corefiber around 1.5 microns is about 3.5 times higher than that of an SMF28fiber. If the gyroscope utilizes the Crystal Fiber air-core PBFmentioned above instead, which is representative of the current state ofthe art, its effective scattering coefficient α_(PBF)=ηα_(silica) isabout 3.5 times larger. This fiber has an NA around 0.12 and a modegroup index close to unity (n≈1), so its solid angle is Ω≈0.045. Thebackscattering noise provided by Equation (3) is therefore δφ≈0.43microradians, which is still very small. Thus, even thoughbackscattering is larger in current air-core fibers than in conventionalfibers, the coherence length of an Er-doped SFS can be short enough tobring the backcattering noise down to a negligible level, and therotation sensitivity is approximately the same with either type offiber.

For air-core PBFs with ultra-low backscattering, the backscatteringnoise will be reduced in proportion to √η, as expressed by Equation (3).For example, for a PBF with η=0.002 and Ω=0.045, Equation (3) predictsδφ≈0.01 microradians. The implication is that with such a fiber, thebackscattered signal is so weak that the gyroscope could now be operatedwith light of much longer coherence length and still have a reasonablylow noise. In the above example, a coherence length as large asL_(c)=2.2 meters (linewidth of 95 MHz) will still produce a noise ofonly 1 microradians, which is low enough for many applications. Thus,rather than using a broadband source, a standard semiconductor laser canbe used, thereby providing significant advantages over broadbandsources, including but not limited to a far greater wavelength stabilityand thus scale-factor stability, and a lower cost, all beneficial stepsfor inertial-navigation FOGs.

With regard to the Shupe effect, the air-core fiber provides asubstantial reduction in thermal sensitivity. When the temperature ofthe sensing coil varies asymmetrically with respect to the coil'smid-point, the two counter-propagating signals sample the resultingthermal phase change at different times, which results in an undesirabledifferential phase shift in the gyroscope output. This effect is reducedin practical gyros by wrapping the coil fiber in special ways, such asquadrupole winding. However, this solution is not perfect, and residualdrifts due to time-varying temperature gradients can be observed inhigh-sensitivity FOGs. When the sensing coil is made with an air-corefiber, because the mode travels mostly in air, whose index depends muchmore weakly on temperature than the index of silica, the Shupe effect issubstantially reduced. For example, in the air-core fiber used in thisexample, the Shupe effect is reduced by a factor of about 3.6 timescompared to a conventional fiber. An FOG made with this fiber wouldtherefore exhibit a thermal drift only about 28% as large as aconventional FOG. Even greater reductions in the Shupe constant, by upto a factor of 11, are obtained with straightforward improvements in thefiber jacket thickness and material. This reduction in Shupe effectadvantageously translates into a greater long-term stability andsimplified packaging designs.

FIG. 7 schematically illustrates an experimental configuration of anall-fiber air-core PBF gyroscope 705 in accordance with certainembodiments described herein used to verify various aspects,particularly that the noise is of comparable magnitude as in aconventional fiber gyroscope. The light source 716 comprised acommercial Er-doped fiber amplifier 708, which produced amplifiedspontaneous emission centered at 1544 nanometers with a calculatedbandwidth of about 7.2 nanometers. This light from the fiber amplifier708 was coupled through an optical isolator 710 and a power attenuator712 into a 3-dB fiber coupler 730 (to provide a reciprocal output port),then into a fiber-pigtailed LiNbO₃ integrated-optic circuit (IOC) 740that consisted of a polarizer 742 followed by a 3-dB splitter 744 and anelectro-optic modulator 746. The output fiber pigtails 748 of the IOC740 were butt-coupled to a 235-meter length of coiled HC-1550-02air-core fiber 750 manufactured by Blaze Photonics (now Crystal FibreA/S of Birkerod, Denmark). This fiber 750 was quadrupolar-wound on an8.2-centimeter mandrel and placed on a rotation stage 752. At thebutt-coupling junctions, the ends of the air-core fiber 750 were cleavedat normal incidence while the ends of the pigtails 748 of the IOC 740were cleaved at an angle to eliminate unwanted Fresnel reflections. Apolarization controller 760 was placed on one of the pigtails 748 insidethe loop to maximize the visibility of the return signal. The outputsignal from the reciprocal port of the 3-dB fiber coupler 730 wasdetected with a photo-detector 770. The attenuator 712 was adjusted sothat the detected output power was same in all measurements (−20 dBm).Both the fundamental frequency (f) and the second-harmonic frequency(2f) of the detected electrical signal were extracted with a lock-inamplifier 46 and recorded on a computer. All results shown were obtainedwith a lock-in integration time of 1 second.

FIG. 8A shows a scanning electron micrograph of a cross-section of theair-core fiber 750. FIG. 8B illustrates the measured transmissionspectrum of the air-core fiber 750 and the source spectrum from thelight source 716. The transmission spectrum includes the coupling lossbetween the air-core PBF 750 and the fiber pigtails 748 of the IOC 740.The fiber 750 is almost single moded in its transmission range (about1490-1660 nanometers). The highest measured transmission value, near themiddle of the PBF bandgap (around 1545 nanometers) is about −10 dB.Based on the minimum fiber loss of about 19 dB/kilometer from themanufacturer, the 235-meter length of air-core fiber 750 accounts forabout 4.5 dB of the 10-dB transmission loss. The rest can be assigned toa about 2.7-dB loss at each of the two butt junctions. The fundamentalmode overlap with the silica portions (parameter η) calculated for thisfiber is a few percent. The measured birefringence was approximately6×10⁻⁵ . From the value of the group index for the fundamental mode ofthe 5-meter standard-fiber pigtails (1.44) and of the 235-meter PBFfiber (1.04, calculated for an ideal air-core fiber with the same coreradius), the proper frequency of the Sagnac loop was calculated to bef₀≈596 kHz. Since this calculated value is approximate, it was alsomeasured by observing the evolution of the 2f signal when the gyroscopewas at rest as the modulation frequency was increased up to 700 kHz. The2f signal amplitude exhibited the frequency response of a Sagnacinterferometer, i.e., a raised sine. A fit to this dependence gave ameasured proper frequency of 592 kHz, in agreement with the calculatedvalue. The air-core gyroscope was modulated at f=600 kHz, with amodulation depth of π/4, close to the optimum value for maximumsensitivity.

The gyroscope was tested by placing it on a rotation table with acalibrated rotation rate and measuring the amplitudes of the f and 2fsignals as a function of rotation rate. FIGS. 9A and 9B show typicaloscilloscope traces of these signals, with and without rotation,respectively. In the absence of rotation (FIG. 9A), the output containsonly even harmonics of f (mostly 2f). When the coil is rotated (FIG.9B), a component at f appears. The output signal at f was measured to beproportional to the rotation rate up to the maximum tested rotation rateof 12°/second. The proportionality constant (of the order of 10-20 mV/°s, depending on experimental conditions) was used to calibrate the noiselevel of the gyroscope in units of rotation rate. A trace of the foutput signal recorded over one hour while the gyroscope was at rest isplotted in FIG. 10. Again, the vertical axis in this curve wascalibrated not from the knowledge of the gyroscope scale factor, but byusing the rotation table calibration. Taking the peak-to-peak noise inthis curve to be 3σ, where σ is the noise standard deviation, theseresults show that 3σ≈8°/hour. Thus the minimum detectable rotation rate(one standard deviation) is Ω_(min)≈2.7°/hour, or about ⅙th of Earthrate. The long-term drift observed in FIG. 10 is about 2°/hour. Thescale factor for this gyro, calculated from the coil diameter and fiberlength, is F=0.26 s. The short-term phase noise of the interferometerinferred from the measured minimum detectable rotation rate is thereforeΩ_(min)F≈3.3 microradians, which is reasonably low for an opticalinterferometer.

To compare this performance to that of a conventional fiber gyroscope,the air-core PBF coil was replaced with a coil of 200 meters of standardPM fiber. This second coil was also quadrupole-wound, on a mandrel2.8-centimeters in diameter. The ends of the coil fiber were spliced tothe IOC pigtails 748. The proper frequency of this Sagnac loop wasmeasured to be around 500 kHz (calculated value of 513 Hz) and thecalculated scale factor was F=0.076 s. The rest of the sensor, includingthe IOC 740, photodetector 770, and detection electronics, were the sameas for the air-core gyroscope described above. The minimum detectablerotation rate of this standard gyroscope was measured to be 7°/hour (1−σshort-term noise) and its long-term drift was 3°/hour. From this minimumdetectable signal and the scale factor, a phase noise of 2.6microradians can be inferred. This value shows the important result thatthe phase noise in the air-core fiber gyroscope is comparable to that ofa standard-fiber gyroscope of comparable length using the sameconfiguration, detection, and electronics.

The three main contributions to the short-term noise observed in thePM-fiber gyroscope are shot noise, d, excess noise due to the broadbandspectrum of the light source, and thermal noise in the detector. Shotnoise is typically negligible. At lower detected power, the dominantsource of noise is thermal detector noise. At higher detected power, thedominant noise is excess noise. Under the conditions of the experimentalair-core fiber gyroscope described above, the detected power was around10 microwatts, and the observed noise (2.6 microradians) originatesalmost entirely from excess noise.

For the air-core FOG, since the detector and detected power were thesame, the shot noise contribution is also 0.4 microradians. Thebackscattering noise, also calculated earlier, is about 0.4 microradians(assuming that the Rayleigh backscattering coefficient of the air-corefiber is 1.12×10⁻⁷ m⁻¹, or 3.5 times stronger than that of an SMF28fiber). Since both test gyroscopes used the same electronics anddetected power levels, the contribution of electronic noise must be thesame as in the PM-fiber gyroscope, namely about 2 microradians.

In addition, in the air-core FOG, the backreflections at the twobutt-coupled junctions were also a possible source of noise. The twobutt-junctions form a spurious Michelson interferometer that is probedwith light of coherence length much shorter than the path mismatchbetween the Michelson's arms, so this interferometer only adds intensitynoise. The magnitude of this noise can be estimated using the knowledgeof the power reflection at the end of an air-core fiber, which is muchweaker than for a solid-core fiber but not zero. Such estimates showthat the power reflection at the end of in this air-core fiber is about2×10⁻⁶. Scaling by this value, the phase noise due to these twoincoherent reflections is estimated to be roughly of the order of 1microradians. Note that this contribution can be eliminated by anglingthe ends of the air-core fiber as well.

The above calculated noise levels are consistent with the measurements:the sum of all four contributions (0.4+0.4+2.3+1=4.1) is comparable tothe measured value of 3.3 microradians. This agreement gives credence tothe estimated values of the various noise contributions, to the assumedvalue of the air-core fiber's Rayleigh scattering coefficient, and tothe conclusion that in both gyroscopes most of the noise arises from acommon electronic origin.

These measurements also allow an upper-bound value to be placed on theRayleigh scattering of the air-core fiber. If the observed phase noise(3.3 microradians) is assumed to be entirely due to fiber scattering,Equation (3) can be used to show that the fiber's Rayleigh scatteringcoefficient would be as high as 6.6×10⁻⁶ m⁻¹, or about 200 times higherthan for an SMF28 fiber. The above assignment of the various noisesources strongly suggest that this value is unreasonably high, and thata value of 1.12×10⁻⁷ m⁻¹ is much more consistent with theseobservations.

EXAMPLE 2

This example models quantitatively the dependence of thefundamental-mode phase on temperature in an air-core fiber, andvalidates these predictions by comparing them to values measured inactual air-core fibers. The metric cited in this example is the relativechange in phase S, referred to as the phase thermal constant, and givenby Equation (4):

$\begin{matrix}{S = {\frac{1}{\phi}\frac{\mathbb{d}\phi}{\mathbb{d}T}}} & (4)\end{matrix}$where φ is the phase accumulated by the fundamental mode through thefiber and T is the fiber temperature. With an interferometric technique,two air-core PBFs from different manufacturers were tested and foundthat their thermal constant is in the range of 1.5 to 3.2 parts permillion (ppm) per degree Celsius, or 2.5-5.2 times lower than themeasured value of a conventional SMF28 fiber (S=7.9 ppm/° C.). Each ofthese values falls within 20% of the corresponding predicted number,which lends credence to the theoretical model and to the measurementcalibration. This study shows that the reason for this reduction is dueto a drastic reduction in the dependence of the mode effective index ontemperature. The residual value of the thermal constant arises fromlength expansion of the fiber, which is only marginally reduced in anair-core fiber. Modeling shows that with fiber jacket improvements, thiscontribution can be further reduced by a factor of about 2. Even withoutthis further improvement, the phase thermal constant of current air-corefiber is as much as about 5 times smaller than in a conventional fiber,which can bring forth a significant improvement in the FOG and otherphase-sensitive systems.

The phase thermal constants S of an air-guided photonic-bandgap fiberand a conventional index-guided fiber are quantified in the followingtheoretical model. The total phase φ accumulated by the fundamental modeas it propagates through a fiber of length L is expressed by Equation(5):

$\begin{matrix}{\phi = \frac{2\pi\; n_{eff}L}{\lambda}} & (5)\end{matrix}$where L is the fiber length, n_(eff) the mode effective index, and λ thewavelength of the signal in vacuum. Inserting Equation (5) into Equation(4) yields the expression for the phase sensitivity per unit length andper degree of temperature change of the fiber, as expressed by Equation(6):

$\begin{matrix}{S = {{\frac{1}{\phi}\frac{\mathbb{d}\phi}{\mathbb{d}T}} = {{{\frac{1}{L}\frac{\mathbb{d}L}{\mathbb{d}T}} + {\frac{1}{n_{eff}}\frac{\mathbb{d}n_{eff}}{\mathbb{d}T}}} = {S_{L} + {S_{n}.}}}}} & (6)\end{matrix}$where dφ/dT is the derivative of the phase delay with respect totemperature. S is the sum of two terms: the relative variation in fiberlength per degree of temperature change (hereafter called S_(L)), andthe relative variation in the mode effective index per degree oftemperature change (hereafter called S_(n)).

If the temperature change from equilibrium is ΔT(t,l) at time t and inan element of fiber length dl located a distance l from one end of thefiber, the total phase change in a length L of the fiber is expressed byEquation (7):

$\begin{matrix}{{\Delta\phi} = {\frac{2\pi\; n_{eff}}{\lambda}S{\int_{0}^{L}{\Delta\;{T\left( {{t - {l/v}},l} \right)}{\mathbb{d}l}}}}} & (7)\end{matrix}$where v=c/n_(eff) and c is the velocity of light in vacuum. Equation (7)shows that S is a relevant parameter to characterize the phasesensitivity to temperature in, for instance, a Mach-Zehnderinterferometer, since the total phase change is proportional to S.

Similar expressions apply to other interferometers. For example, for aSagnac interferometer the corresponding phase change is given byEquation (8):

$\begin{matrix}{{\Delta\phi} = {\frac{2\pi\; n_{eff}}{\lambda}S{\int_{0}^{L}{\left\lbrack {{\Delta\;{T\left( {{t - {l/v}},l} \right)}} - {\Delta\;{T\left( {{t - {L/v} + {L/v} + {l/v}},l} \right)}}} \right\rbrack{\mathbb{d}l}}}}} & (8)\end{matrix}$As expected, Δφ is proportional to S, and S is again the relevantmetric. The temperature sensitivity of a Sagnac interferometer,expressed by Equation (8), can be reduced by minimizing the integralthrough proper fiber winding, and/or by designing the fiber structure tominimize S.

Because the thermal expansion coefficient of the fiber jacket (usually apolymer) is typically two orders of magnitude larger than that ofsilica, expansion of the jacket stretches the fiber, and the fiberlength change caused by jacket expansion is the dominant contribution toS_(L). The index term S_(n) is the sum of three effects. The first oneis the transverse thermal expansion of the fiber, which modifies thecore radius and the photonic-crystal dimensions, and thus the modeeffective index. The second effect is the strains that develop in thefiber as a result of thermal expansion; these strains alter theeffective index through the elasto-optic effect. The third effect is thechange in material indices induced by the fiber temperature change(thermo-optic effect).

To determine S_(n) and S_(L), the thermo-mechanical properties of thefiber are modeled by assuming that the fiber temperature is changeduniformly from T₀ to T₀+dT, and calculating the fiber length and theeffective index of the fundamental mode at both temperatures, fromwhich, with Equation (6), S_(n), S_(L), and S can be calculated. FIG. 11schematically illustrates a fiber 800 in accordance with certainembodiments described herein. The fiber 800 is assumed to havecylindrical symmetry and all its properties are assumed to be invariantalong its length, so S is computed in a cylindrical coordinate system.As shown in FIG. 11, the fiber 800 is modeled as a structure withmultiple circular layers: a core 810 (e.g., doped silica in aconventional fiber, hollow in a PBF, and shown as filled with air inFIG. 11) of radius α₀, an inner cladding 820 (e.g., silica in aconventional fiber, a silica-air honeycomb in a PBF) generallysurrounding the core 810, an outer cladding 830 (generally pure silica)of radius α_(M) and generally surrounding the inner cladding 820, and ajacket 840 (often an acrylate) generally surrounding the outer cladding830. Each layer is assumed to remain in contact and in mechanicalequilibrium with the neighboring layers, i.e., the radial stress and theradial deformation are continuous across fiber layer boundaries. Incertain embodiments, the core 810 has an outer diameter in a rangebetween about 9 microns and about 12 microns. In certain embodiments,the inner cladding 820 has an outer diameter in a range between about 65microns and about 72 microns. In certain embodiments, the outer cladding830 has an outer diameter in a range between about 110 microns and about200 microns. In certain embodiments, the jacket 840 has an outerdiameter in a range between about 200 microns and about 300 microns.

Each layer is characterized by a certain elastic modulus E, Poisson'sratio ν, and thermal expansion coefficient α. The photonic-crystalcladding 820 is an exception in that it is not a homogeneous materialbut it behaves mechanically like a honeycomb. The implications are that(1) in a transverse direction the honeycomb can be squeezed much moreeasily than a solid, which means that it has a high transverse Poisson'sratio, and (2) in the longitudinal direction, it behaves like membranesof silica with a total area (1−η)A_(h), where A_(h) is the total crosssection of the honeycomb. The elastic modulus and Poisson's ratio of ahoneycomb are thus function of the air filling ratio η. For an hexagonalpattern of air holes in silica, they are given by Equation (9):

$\begin{matrix}{{E_{T} = {\frac{3}{2}\left( {1 - \eta} \right)^{3}E_{0}}}{E_{L} = {\left( {1 - \eta} \right)E_{0}}}{v_{T} = 1}{v_{L} = v_{0}}} & (9)\end{matrix}$where E_(T) and E_(L) are the transverse and longitudinal Young'smodulus of the silica-air honeycomb, respectively, E₀ is the Young'smodulus of silica, ν_(T) and ν_(L) are the transverse and longitudinalPoisson's ratios of the honeycomb material, respectively, and ν₀ is thePoisson's ratio of silica. The values used for these parameters in thesimulations presented here were calculated from Equation (9) and arelisted in Table 1. Comparison to a simpler model in which the innercladding is approximated by solid silica indicates that the effect ofthe honeycomb is to increase S_(L) by about 10-30%. The reason is thatin a honeycomb offers a lower resistance to the pull exerted by thehigher thermal expansion jacket than solid silica, thus the fiber lengthexpansion is increased (larger S_(L)). The effect of the honeycomb isthus small but not negligible. Table 1 also lists the values of theparameters used in the simulations for the other fiber layers.

TABLE 1 Coefficient Silica Air-Silica Honeycomb Acrylate Thermalexpansion 0.55 0.55 80 coefficient α (10⁻⁶/K) Poisson's ratio ν 0.17Transverse: 1 0.37 Longitudinal: 0.17 Young's modulus E 72.45Transverse: 108.7 (1-η)³ 0.5 (GPa) Longitudinal: 72.45 (1-η)

The local deformation vector u(r) at the point r=[r, θ, z] is given byEquation (10):u(r)=[u _(r)(r) 0 u _(z)(z)]  (10)Only the diagonal components of the strain tensor ε are non-zero:

$\begin{matrix}{ɛ = \begin{bmatrix}{ɛ_{rr} = \frac{\partial u_{r}}{\partial r}} & {ɛ_{\theta\theta} = \frac{u_{r}}{r}} & {ɛ_{zz} = \frac{\partial u_{z}}{\partial z}}\end{bmatrix}} & (11)\end{matrix}$

Hooke's law is used to relate the stress tensor σ and strain tensor εalong with the effect of a temperature change ΔT:ε=s : σ+αΔT  (12)where s is the fourth-order compliance tensor, α is the thermalexpansion tensor, which also only has diagonal terms, and : denotes thetensor product.

The deformation field u_(z) does not vary with r and for a long fiber,it varies linearly with z, such that it is of the form:u _(z)(z)=Cz  (13)where C is a constant and the z origin is chosen in the middle of thefiber. Because u_(z)(z) is continuous at each interface between layers,C has the same value for all layers. Since the temperature is assumeduniform across the fiber, Equations (12) and (13) imply that ε_(zz) andσ_(zz) are independent of r and only functions of z. Furthermore, u_(r)satisfies the admissibility condition:

$\begin{matrix}{{\frac{\partial^{2}u_{r}}{\partial r^{2}} + {\frac{1}{r}\frac{\partial u_{r}}{\partial r}} - \frac{u_{r}}{r^{2}}} = 0} & (14)\end{matrix}$whose solution is:

$\begin{matrix}{{u_{r}(r)} = {{A\; r} + \frac{B}{r}}} & (15)\end{matrix}$where A and B are constants specific to each layer. The coefficients A,B, and C are solved for by imposing the following boundary conditionsand making use of Hooke's law: (i) continuity of u_(r)(r) across allinner layer boundaries; (ii) continuity of σ_(rr)(r) across all innerlayer boundaries; (iii) σ_(rr)(r)=0 at r=a₀, where a₀ is the fiber coreradius; (iv) σ_(rr)(r)=0 at r=a_(M), where a_(M) is the outer radius ofthe fiber; and (v) mechanical equilibrium on the fiber end faces, whichimposes:

$\begin{matrix}{{\int_{a_{0}}^{a_{M}}{\int_{0}^{2\pi}{{\sigma_{zz}\left( {r,\theta,{z = {{\pm L}/2}}} \right)}{\mathbb{d}r}{\mathbb{d}\theta}}}} = 0} & (16)\end{matrix}$A matrix method can be used to determine A, B, and C, and thus u_(r)(r)and u_(z)(z). Equation (11) then yields the strains, includingε_(zz)=S_(L)ΔT.

To illustrate the kinds of predictions this model provides, FIG. 12shows the radial deformation as a function of distance from the fibercenter calculated for the Crystal Fibre PBF with the physical parameterslisted in Table 2. Over the honeycomb structure (inner radius of about 5microns and outer radius of about 33.5 microns) and the silica outercladding (inner radius of about 33.5 microns and outer radius of about92.5 microns), the radial deformation remains small compared to thedeformation of the acrylate jacket (inner radius of about 92.5 micronsand outer radius of about 135 microns), which is consistent with thedifferences in the thermal expansion coefficient and stiffness of thematerials. The low-thermal expansion and stiff silica experiences a muchweaker deformation than the high-thermal expansion and soft acrylate.Since the radial strain is the derivative of the radial deformation, theinner cladding honeycomb is under compressive strain, and it relaxes thestrain over the structure by absorbing some of the deformation (theradial deformation decreases by a factor of 4 over the honeycombstructure) due to its very small transverse Young modulus.

TABLE 2 Parameter SMF28 Fiber Crystal Fibre Blaze Photonics Air-fillingratio Not applicable >90% >90% Core diameter 8.2 μm 10 μm 10.9 μmHoneycomb diameter Not applicable 67 μm 70 μm Cladding diameter 125 μm185 μm 120 μm Jacket diameter 250 μm 270 μm 220 μm Heated length offiber 210 cm 226 cm 210 cm

Once the strain distributions are known, computation of S_(n) isstraightforward. In a first step, from the radial strain distribution,the change in the dimension of each layer across the fiber cross-sectionis calculated. In a second step, from the total strain distribution, thechange in refractive indices of each layer due to the elasto-opticeffect is calculated. In a third step, the change in material indicesinduced by the temperature change (thermo-optic effect), which isindependent of the strains and the easiest to evaluate, is calculated.These three contributions (change in index profile, core radius, andmaterials' indices) are then combined to obtain the refractive indexprofile of the fiber at T=T₀+ΔT. This new profile is then imported intoan appropriate code to calculate various optical properties of thestructure (see, e.g., V. Dangui, M. J. F. Digonnet and G. S. Kino, “Afast and accurate numerical tool to model the mode properties ofphotonic-bandgap fibers,” Optical Fiber Conference Technical Digest(2005)), such as the effective index of the fundamental mode at thistemperature. The code is also used to compute the mode effective indexof the unperturbed fiber, i.e., at temperature T₀. These two values ofthe effective index are used in Equation (6) to compute S_(n). Thiscalculation assumes that all parameters change linearly withtemperature, which is reasonable for small temperature excursions.

FIG. 13 shows the dependence of S_(L), S_(n), and S on the core radius Rpredicted by this model for a fiber with a cladding air-hole radiusρ=0.495Λ, an outer cladding radius of 92.5 μm, and an acrylate jacket ofthickness 42.5 μm (parameters of the Crystal Fibre PBF). The signalwavelength was λ=0.5Λ, close to the middle of the fiber bandgap, with acenter-to-center distance between adjacent hollow regions in the innercladding of Λ=3 μm. The values of S, S_(L), and S_(n) calculated for anSMF28 fiber (see parameters in Table 2) at λ=1.5 μm are also indicatedfor comparison. S_(L) is almost independent of core radius and is thedominant term. The situation is reversed from a conventional SMF28fiber, for which S_(n) is significantly larger than S_(L). Note alsothat S_(L) is sensibly the same for the air-core and the SMF28 fibers.The physical reason is that S_(L) quantifies linear expansion of thefiber, which is similar in both fibers. S_(L) is actually a little lowerfor the air-core fiber because of the increased relative area of silicain the outer cladding compared to the acrylate in the jacket for thisparticular fiber. Therefore, the PBF has a lower overall thermalexpansion than the SMF28 fiber. For the air-core fiber, the index termS_(n) generally decreases slowly with increasing core radius, except forprominent local peaks in the ranges of 1.1Λ-1.25Λ and 1.45Λ-1.65Λ, whereS_(n) and S increase by as much as a factor of two. These rangescoincide precisely with the regions when surface modes occur(highlighted in gray in FIG. 13). The reason is that for the core radiithat support surface modes, a significantly larger fraction of thefundamental mode energy is contained in the dielectric portions of thefiber, and the phase is more sensitive to temperature. This resultpoints out yet another reason why surface modes should be avoided.Outside of these surface-mode regions, the total phase thermal constantS=S_(L)+S_(n) varies weakly with core radius. The lowest S value in thesingle-mode range (R<˜1.1Λ) occurs for R≈1.05Λ and is equal to about1.68 ppm/° C., which is 4.9 times smaller than for an SMF28 fiber.

Since in an air-core fiber most of the contribution to S comes from thelength term S_(L), the more complex index term S_(n) can be neglectedand it is worth developing a simple model to evaluate S_(L) (and thusS). The value of S_(L) can be approximated to a good accuracy, whilegaining some physical insight for the effects of the various parameters,by ignoring the radial terms and utilizing the condition that the totalforce exerted on the fiber in the z direction is zero. Using thenotation in FIG. 11, this total force can be expressed as:σ_(zz,h) A _(h)+σ_(zz,cl) A _(cl)+σ_(zz,J) A _(J)=0  (17)where the subscripts h, cl, and J stand for honeycomb, outer cladding,and jacket, respectively. The corresponding term for the air core iszero and is thus absent from Equation (17). Substituting Equation (12)into Equation (17) while neglecting the transverse terms, which aresmall because the fiber radius is small compared to the fiber length,the following approximate expression is obtained for S_(L):

$\begin{matrix}{S_{L} = {\frac{\Delta\; L}{L\;\Delta\; T} = {\frac{ɛ_{zz}}{\Delta\; T} \approx \frac{{A_{h}E_{h}\alpha_{h}} + {A_{cl}E_{cl}a_{cl}} + {A_{J}E_{J}\alpha_{J}}}{{A_{h}E_{h}} + {A_{cl}E_{cl}} + {A_{J}E_{J}}}}}} & (18)\end{matrix}$

As can be seen from Table 1, the jacket expansion term A_(J)E_(J)α_(J)and the outer cladding expansion term A_(cl)E_(cl)α_(cl) are comparablein size and much larger than the honeycomb term A_(h)E_(h)α_(h), whichcan be neglected. In the denominator, the main term is the restoringforce A_(cl)E_(cl) due to the outer cladding, which is much larger thanthe force from the jacket or the honeycomb. Hence, Equation (16) can bewell approximated by:

$\begin{matrix}{{{S_{L} \approx \frac{{A_{cl}E_{cl}\alpha_{cl}} + {A_{J}E_{J}\alpha_{J}}}{A_{cl}E_{cl}}} = {\alpha_{cl} + \frac{A_{J}}{A_{cl}}}}{\frac{E_{J}}{E_{cl}}\alpha_{J}}} & (19)\end{matrix}$This simple expression shows that S_(L) can be lowered by making thearea of the outer cladding A_(cl) as large as possible relative to thejacket area A_(J), and by using a jacket material with a low thermalexpansion. This approximate model turns out to be quite accurate. Incertain embodiments described herein, the area of the outer claddingA_(c), the area of the jacket A_(J), the Young's modulus of the outercladding E_(cl), the Young's modulus of the jacket E_(J), and thecoefficient of thermal expansion of the outer cladding α_(cl), and thecoefficient of thermal expansion of the jacket α_(J), are selected suchthat the quantity

$\frac{A_{J}}{A_{cl}}\frac{E_{J}}{E_{cl}}\frac{\alpha_{J}}{\alpha_{cl}}$is less than or equal to 2.5, while in certain other embodiments, thisquantity is less than 1.

The parameter S was measured for two PBF fibers, namely the AIR-10-1550fiber manufactured by Crystal Fibre A/S and the HC-1550-02 fiber fromBlaze Photonics (now Crystal Fibre A/S). SEM photographs of the fibers'cross-sections are shown in FIGS. 14A and 14B, respectively.Measurements were carried out using the conventional Michelson fiberinterferometer schematically illustrated in FIG. 15A. The signal sourcewas a 1546-nm DFB laser with a linewidth of a few MHz. The air-corefiber was either spliced (Blaze Photonics fiber) or butt-coupled(Crystal Fiber fiber) to one of the ports of a 3-dB coupler (SMF28fiber) to form the “sensing” arm of the interferometer. The far end ofthe PBF was similarly coupled to a fiber-pigtailed Faraday rotatormirror (FRM) to reflect the signal back through the fiber and thuseliminating polarization fluctuations in the return signal due tovariations in the fiber birefringence. Most of the PBF was attached toan aluminum block placed on a heating plate, and the fiber/blockassembly was covered with a styrofoam thermal shield (shownschematically in FIG. 15A) to maintain the fiber temperature as uniformas possible and to reduce temperature fluctuations due to air currentsin the room. The temperature just above the surface of the block wasmeasured with a thermocouple (e.g., output 1 mV/° C.).

The second (reference) arm of the interferometer consisted of a shorterlength of SMF28 fiber splice to a second FRM. Together with the non-PBFportion of the sensing arm, this entire arm was placed in a secondthermal shield (shown schematically in FIG. 15A), mostly to reduce theamount of heating by the nearby heater of both the reference fiber andthe non-PBF portion of the sensing arm. With this arrangement, when theheater was turned on the PBF was the only portion of the interferometerthat was significantly heated.

To measure S, the temperature of the PBF was raised to around 70° C.,then the heater was turned off and as the PBF temperature slowlydropped, both the output power of the interferometer and the fibertemperature were measured over time and recorded in a computer. Duringthe measurement time window (typically tens of minutes), the phase inthe PBF arm decreased and passed many times (e.g., 50-200) through 2π,so that the power at the interferometer output exhibited many fringes,as illustrated in the typical experimental curves of FIG. 16. The phasethermal constant S was calculated from the measured number of fringesoccurring in a given time interval using:

$\begin{matrix}{S = {\frac{\Delta\phi}{4\pi\; n_{eff}L\;\Delta\;{T/\lambda}} \approx \frac{N_{fringes}}{2n_{eff}L\;\Delta\;{T/\lambda}}}} & (20)\end{matrix}$where L is the length of fiber under test, ΔT is the temperature changeoccurring during the measurement interval, and N_(fringes) is the numberof fringes, which are illustrated in FIGS. 16A and 16B.

This approximation is justified because the temperature drop was slowenough that the PBF temperature was uniform at all times, yet fastenough that random phase variations in the rest of the interferometerwere negligible compared to the phase variations in the PBF. To verifythis last point, the inherent temperature stability of theinterferometer was measured by disconnecting the PBF and reconnectingthe fiber ends of the sensing arm with a short length of SMF28 fiber, asillustrated in FIG. 15B. In a first stability test, the interferometeroutput was recorded while the entire interferometer temperature was atequilibrium room temperature. Over a period of about 30 minutes, theenclosure temperature was found to vary by ±1° C. and the output powervaried by about one fringe only. This test showed that theinterferometer was more than stable enough to measure phase shifts oftens of fringes.

In a second test, the PBF enclosure was heated to around 70° C., thenthe heater was turned off and the interferometer output was recorded asthe heater slowly cooled down. This time a larger number of fringes wereobserved, which indicated that a little heat from the heater reachedthrough the interferometer shield and induced a differential temperaturechange in the two arms. The output power varied by about 12 fringeswhile the enclosure temperature dropped about 18° C. Consequently, whenmeasuring S with the setup of FIG. 15A, residual heating of the non-PBFportion of the interferometer introduces an error of about 12 fringes.For this error to be small compared to the fringe count due to thechange in the PBF temperature, this fringe count should be much largerthan the error, e.g., 100 or more. This condition was met by using asufficiently long PBF. For the value of S≈2 ppm/° C. predicted for a PBF(e.g., FIG. 13), Equation (18) predicts that the length required toobtain 100 fringes of phase shift for a ΔT of 18° C. is L≈1 meter. Thelength of PBF used in the measurements described herein was therefore ofthis order (about 2 meters, as shown in Table 2).

As a point of comparison, the thermal constant of a conventionalsolid-core fiber was measured by replacing the PBF in the experimentalsetup of FIG. 15A by a 210-cm length of SMF28 fiber. The measured valuewas S=7.9 ppm/° C., in excellent agreement with the value of 8.2 ppm/°C. predicted by the model using the parameter values of Tables 1 and 2.This value is the sum of S_(L)=2.3 ppm/° C and S_(n)=5.9 ppm/° C., i.e.,the index contribution is 2.6 times larger than the length expansioncontribution. These values are summarized in Table 3. The closeagreement between measured and calculated values gives credence to boththe model and the interferometer calibration.

TABLE 3 Crystal Blaze Bragg Fiber SMF28 Fibre Photonics Fiber S,measured (ppm/° C.) 7.9 1.5 ± 0.9 2.2 ± 0.7 S, predicted (ppm/° C.) 8.21.42 2.62 1.45 S_(L) (ppm/° C.) 2.3 1.36 2.57 1.15 S_(n) (ppm/° C.) 5.90.06 0.05 0.30

The value of S was then measured for the two air-core PBFs. A typicalexperimental result is shown in FIGS. 16A and 16B. The value of Sinferred for each fiber from such measurement and Equation (18) islisted in Table 3, along with the calculated values of S, S_(n), andS_(L). The S values measured for the two PBFs are fairly similar, in therange of 1.5 to 2.2 ppm/° C. As predicted, the air-core fiber guidancemechanism results in a sizable decrease in the sensitivity of the phasedelay on temperature. This reduction is as much as a mean factor of 5.26(measured) or 5.79 (predicted) for the Crystal Fibre PBF. Thecorresponding figures for the Blaze Photonics fiber are 3.6 (measured)and 3.14 (predicted). Again, the theoretical and measured values agreewell. The Crystal Fibre fiber exhibits a lower thermal expansioncontribution than the Blaze Photonics fiber because it has a larger areaof silica cladding relative to the jacket area, as expressed by Equation(17). These reductions in S result mostly from a decrease in S_(n) by afactor of about 100, as well as a 15%-45% reduction in S_(L); aspredicted by theory, in a PBF S is determined overwhelmingly by S_(L),which depends only on the change in fiber length. The conclusion is thatcurrent air-core fibers are substantially less temperature sensitivitythat conventional fibers, by a factor large enough (e.g., 3.6-5.3) thatit will translate into a significant stability improvement in fibersensors and other phase-sensitive fiber systems.

Even smaller values of S can be obtained with improved PBF designs.Since in a PBF, S_(L) is the main contribution to S, to further reduceS, the value of S_(L) can be reduced. This term arises from the thermalchange in the fiber length, which is driven by both the thermalexpansion coefficient and the stiffness of (i) the honeycomb cladding(e.g., silica and air), (ii) the outer cladding (e.g., silica), and(iii) the jacket (e.g., a polymer). Because polymers have a much higherthermal expansion coefficient than silica, as the temperature isincreased the jacket expands more than the fiber, and thus it pulls onthe fiber and increases its length more than if the fiber wasunjacketed. The jacket is therefore generally the dominant contributionto S_(L). Consequently, a thinner jacket will result in a smaller S_(L),the lowest value being achieved for an unjacketed fiber. Furthermore,everything else being the same a softer jacket (lower Young modulus)will stretch the fiber less effectively and thus yield a lower S_(L). Inaddition, increasing the outer cladding thickness increases the overallstiffness of the fiber structure, thus reducing the expansion of thehoneycomb and reducing S_(L).

These predictions were confirmed by simulating the Blaze Photonics fiberfor various acrylate jacket thicknesses and air filling ratios, as shownin FIG. 17. As the jacket thickness is reduced, S_(L) decreases. In thelimit of zero jacket thickness (bare fiber), S_(L) reaches its lowestlimit, set by the thermal expansion of the silica cladding. For higherair filling ratios, S_(L) is observed to be larger. The reason is thatthe honeycomb then contains less silica, the fiber has a lower overallstiffness, and the jacket expansion is less restrained by the glassstructure, resulting in a larger S_(L) value.

The effect of the jacket material stiffness can be seen in FIG. 18,where the calculated values of S_(L) is graphed (for the same PBF, withan air filling ratio of 90%) for a few standard jacket materials(metals, polymers, and amorphous carbon covered with polyimide). Tosimulate actual fiber jackets, the jacket thickness was taken to be 5 or50 microns for polyimide (as specified), 20 microns for metals, and 20nanometers for amorphous carbon (covered with either 2.5 or 5 microns ofpolyimide). The reference jacket of the actual manufactured PBF was 50microns of acrylate. All metal jackets yield a larger S_(L) than did thereference acrylate jacket (2.57 ppm/° C.). The explanation is that whilemetals have a lower thermal expansion than acrylate (by about one orderof magnitude), their Young modulus is much larger than that of bothacrylate (by 2-3 orders of magnitude) and silica (by a factor of up to3). The silica structure is therefore pulled more effectively by theexpanding metal coating than by the acrylate jacket, and S_(L) islarger. Several jacket materials, however, perform better than acrylate.A thin (20 nanometer) amorphous carbon coating with a 2.5-micronpolyimide jacket over it gives the lowest value, S_(L)=0.67 ppm/° C.(74% reduction), followed by a 5-micron polyimide jacket (S_(L)=0.77ppm/° C., 70% reduction). This is close to the theoretical limit for asilica fiber, which is set by the thermal expansion coefficient ofsilica and is equal to S_(L)=0.55 ppm/° C. The polyimide jacket providesthe lowest value of S_(L) because it is much thinner than an acrylatejacket. For equal thickness, acrylate actually performs better thanpolyimide. But because polyimide is a better water-vapor barrier thanacrylate, a polyimide jacket only a few microns thick is sufficient toeffectively protect the fiber against moisture, which is not true foracrylate. The conclusion is that acrylate unfortunately happens not tobe the best choice of jacket material for thermal performance. Bycoating the PBF with the above carbon-polyimide jacket instead, an S_(L)of only 0.67 ppm/° C., i.e., an S as low as 0.72 ppm/° C., isattainable, which is about 11 times lower than for a conventional fiber.

The effect of the silica outer cladding was also studied by simulatingthe same PBF for increasing cladding thicknesses, assuming a fixed50-micron acrylate jacket and a 90% air-filling ratio. The result isplotted as the solid curve in FIG. 19. As the outer cladding thicknessis increased, S_(L) drops, because a thicker silica cladding betterresists the length increase of the acrylate jacket. This effect isfairly substantial. For example, by doubling the outer claddingthickness from the 50-micron value of the Blaze Photonics fiber to 100microns, S_(L) is reduced by 55%. In the opposite limit of no outercladding (zero thickness), S_(L) jumps up to more than 20 ppm/° C. Thehigh thermal expansion jacket is then pulling only on the silicahoneycomb structure, which has a lower Young modulus due to the airholes and thus offers less resistance to stretching. Using a thick outercladding is therefore an advantageous way of reducing the thermalsensitivity of an air-core fiber. The downside is that the fiber is thenstiffer and can therefore not be wound as tightly, which is adisadvantage in some applications.

The dashed curve in FIG. 19 was generated using the approximate modeldescribed herein. This curve is in very good agreement with the exactresult. Since again S_(L) accounts for more than 90% of the thermalconstant S, this very simple model is a reliable tool to predict thethermal constant of any fiber structure.

Thus, it is possible to reduce the thermal constant below the low valuealready demonstrated in existing air-core fibers by using (i) a jacketas thin as possible; (ii) a soft jacket material; (iii) a large outercladding; and/or (iv) a small air filling ratio (inasmuch as possible).Jacket materials that satisfy (i) and (ii) include, but are not limitedto, polyimide and amorphous carbon covered by a thin layer of polyimide.With a 5-micron polyimide jacket, the Blaze Photonics fiber has athermal constant of S≈0.82 ppm/° C., which is about 3.2 times smallerthan in the current fiber.

In certain embodiments, the phase thermal constant S less than 8parts-per-million per degree Celsius. In certain embodiments, the phasethermal constant S less than 6 parts-per-million per degree Celsius. Incertain embodiments, the phase thermal constant S less than 4parts-per-million per degree Celsius. In certain embodiments, the phasethermal constant S less than 1.4 parts-per-million per degree Celsius.In certain embodiments, the phase thermal constant S less than 1part-per-million per degree Celsius.

The theoretical thermal phase sensitivity to temperature was alsocalculated for a Bragg fiber with a core radius of 2 microns, surroundedby 40 air-silica Bragg reflectors with thicknesses of 0.48 microns(silica) and 0.72 micron (air), with an acrylate jacket thickness of62.5 microns. This fiber exhibits a fundamental mode confined in its aircore with a radius of 1.5 microns. As shown in Table 3, these resultsyielded: S_(L)=1.15 ppm/° C., S_(n)=0.30 ppm/° C., and S=1.45 ppm/° C.Because the fundamental mode in a Bragg fiber travels mostly in air,this value of S is much lower than for a conventional fiber. S iscomparable to the value for a PBF, and the main contribution is againthe lengthening of the fiber.

EXAMPLE 3

FIG. 20 schematically illustrates an example configuration for testing afiber optic gyroscope 905 compatible with certain embodiments describedherein. The sensing coil 910 comprises a Blaze Photonics air-core fiberhaving a length of 235 meters wound in 16 layers on an8-centimeter-diameter spool using quadrupole winding to reduce thethermal and acoustic sensitivities of the coil 910. Each fiber layer inthe coil 910 was bonded to the layer underneath it with a thin epoxycoating, and the outermost layer was also coated with epoxy. This fiberwas essentially single-moded at the signal wavelength of about 1.54microns (the few higher order modes were very lossy). The calculatedscale factor of this coil 910 was 0.255 s. Light from a broadbandEr-doped superfluorescent fiber source (SFS) 920 was isolated byisolator 922, transmitted through a polarization controller (PC₁) 924,and split by a 3-dB fiber coupler before being coupled to afiber-pigtailed LiNbO₃ integrated optical circuit (IOC) 930 comprising apolarizer 932, a 3-dB input-output Y-junction coupler 934, anelectro-optic phase modulator (EO-PM) 936, and a polarization controllerPC₂ 938. The latter was modulated at the loop proper frequency (600 kHz)with a peak-to-peak amplitude of 3.6 rad to maximize the FOGsensitivity.

The Y-coupler 934 splits the input light into two waves that propagatearound the fiber coil 910 in opposite directions. The two ends of theoutput pigtails of the IOC 930, cut at an angle to reduceback-reflections and loss, were butt-coupled to the ends of the PBFsensing coil 910, which were cleaved at 90°. Measured losses were about2-3 dB for each butt-junction, about 4.7 dB for the fiber coil 910, andabout 14 dB round-trip for the IOC 930. The PC₁ 924 was adjusted tomaximize the power entering the interferometer, and the PC₂ was adjustedfor maximum return power at the detector (about 10 μW for 20 mW inputinto the IOC 930). The detector 940 was a low-noise amplified InGaAsphotodiode (available from New Focus of Beckham, Inc. of San Jose,Calif.). Rotating the coil 910 around its main axis induces a phaseshift via the Sagnac effect between the counter-propagating waves, andthe phase shift is proportional to the rotation rate Ω. The modulatedlight signal returning from the coil 910 was detected at the fibercoupler output and analyzed with a lock-in amplifier 950 (100-msintegration time; 24-dB/octave filter slope). This measured signal has alinear dependence on the phase shift, for small rotation rates. Thegyroscope sensitivity was maximized by applying a sinusoidal phasemodulation to the two waves at the loop proper frequency with the EO-PM936.

The short-term noise of this air-core fiber gyroscope 905 was measuredby recording the one-sigma noise level in the return signal as afunction of the square root of the detection bandwidth for integrationtimes ranging from 100 microseconds to 10 seconds. This dependence wasfound to be linear, as expected for a white-noise source, with a slopethat gave the gyroscope's random walk. This measurement was repeated fordifferent signal powers incident on the detector. FIG. 21 shows themeasured dependence of random walk on the signal power, measured at theIOC input. This result indicates clearly that the sensitivity of theair-core fiber gyroscope was limited by two of the three main sources ofnoise typically present in conventional fiber gyroscopes, namelydetector thermal noise for low detected powers (e.g., less than 4 μW)and excess noise from the broadband light source for high detectedpowers (e.g., greater than 4 μW). The third source of noise is shotnoise. At low power, the minimum detectable rotation rate was inverselyproportional to the detected power, while at higher power the noise wasindependent of power. The dashed curve labeled “thermal noise” in FIG.21 is the theoretical contribution of the detector thermal noise,calculated from the detector's noise equivalent power (2.5 pW/√Hz). Thehorizontal dashed curve represents the predicted excess noise calculatedfrom the measured bandwidth of the SFS (2.8 THz). The lowest dashedcurve represents the theoretical shot noise which is inverselyproportional to the square root of the input power. This contribution isnegligible in the air-core gyroscope 905. The sum of these three sourcesof noise is the total expected noise, illustrated by the solid curve inFIG. 21. It is in good agreement with the measured data points. Thiscomparison demonstrates that the performance of this air-core fibergyroscope is limited by excess noise, as it typically is in conventionalFOGs. Importantly, it also shows that the residual back-reflections fromthe butt-coupling junctions between dissimilar fibers, as schematicallyillustrated by FIG. 20, have no impact on the gyroscope's short-termnoise, presumably because of the use of a low-coherence source, andbecause the lengths of the two pigtails differ by more than one sourcecoherence length, which further reduces coherent interaction between theprimary and the reflected waves.

In the excess-noise-limited regime, the random walk of the air-corefiber gyroscope is 0.015 deg/√hr. For an integration time of 80milliseconds, corresponding to a typical detection bandwidth of 1 Hz,the measured minimum detectable phase shift is then 1.1 μrad,corresponding to a minimum detectable rotation rate of 0.9 deg/hr. Thesevalues are very similar to the performance of state-of-the-artcommercial inertial-navigation-grade fiber optic gyroscopes. This resultwas obtained while using the same detected power as in a typicalconventional FOG (e.g., about 10 μW). However, because of the higherpropagation loss of air-core fibers, this power was achieved by using alarger input power than a conventional FOG, namely a few mW, as shown byFIG. 21. This input power could nevertheless easily be reduced by usinga lower noise detector, which would move the crossing point of the twodashed curves of FIG. 21 to a lower power, and would reduce the fiberloss.

The benefits of the air-core fiber gyroscope in certain embodiments liemainly in its improved long-term stability, starting first with itstemperature drift. A thermal transient applied to a Sagnac loop anywherebut at its mid-point induces a differential phase shiftindistinguishable from a rotation-induced phase shift. If thetemperature time derivative is {dot over (T)}(z) in an element of fiberlength dz located a distance z from one end of the coiled fiber, thetotal phase shift error in a fiber of total length L is:

$\begin{matrix}{{\Delta\phi}_{E} = {\frac{2\pi}{\lambda_{0}c}n^{2}S{\int_{0}^{L}{\left( {L - {2z}} \right){\overset{.}{T}(z)}{\mathbb{d}z}}}}} & (21)\end{matrix}$where λ₀ is the wavelength and c is the velocity of light (both invacuum), n is the effective index of the fiber mode, and S is the Shupeconstant. The Shupe constant takes into account both the fiberelongation and the effective index variation with temperature, and isindependent of fiber length. The phase shift error of Equation (21)induces a rotation-like signal Ω_(E) related to Δφ_(E) by:

$\begin{matrix}{{\Delta\phi}_{E} = {\frac{2\pi}{\lambda_{0}c}L\; D\;\Omega_{E}}} & (22)\end{matrix}$where D is the coil diameter.

Substituting Equation (21) into Equation (22), and using a dimensionlessvariable z′=z/L, yields the following expression for the rotation rateerror induced by the transient temperature change {dot over (T)}(z):

$\begin{matrix}{\Omega_{E} = {\frac{n^{2}S\; L}{D}{\int_{0}^{1}{\left( {1 - {2z^{\prime}}} \right){\overset{.}{T}\left( z^{\prime} \right)}{\mathbb{d}z^{\prime}}}}}} & (23)\end{matrix}$

Equation (23) states that the thermal sensitivity of the FOG isproportional not only to the Shupe constant S, but also to n², thesquare of the mode index. Since the air-core fiber has a much smallereffective index (n≈0.99) than does a standard fiber (n≈1.44), as well asa smaller Shupe constant, a dramatic reduction of the thermalsensitivity of the gyroscope is expected by using an air-core fiber. Asshown in Table 3, the Shupe constant for the SMF28 fiber was measured tobe S=7.9 ppm/° C. and the Shupe constant for the Blaze Photonicsair-core fiber was measured to be S=2.2 ppm/° C. Combined with theadditional benefit of this n² dependence, these values suggest that theBlaze Photonics PBF gyroscope should be about 7.6 times less thermallysensitive than the solid-core fiber gyroscope, which constitutes aconsiderable stability improvement.

To verify these predictions experimentally, the output signal of theair-core FOG and of a solid-core FOG were recorded while subjecting thecoils to known temperature cycles. In each case, the sensing coil washeated asymmetrically by exposing one of its sides to warm air from aheat gun, as schematically illustrated by FIG. 20. Prior to thesemeasurements, each gyroscope was carefully calibrated by placing it on arotation table, applying known rotation rates, and measuring the lock-inoutput voltage dependence on rotation rate. The quantity measured duringthe thermal measurements was therefore a rotation error signal Ω_(E)from which Δφ_(E) was inferred using Equation (22).

FIG. 22A shows an example of a measured temporal profile applied to oneside of the air-core fiber gyroscope coil and the measured rotationerror that it induced. Since the rotation error depends on the timederivative of the temperature, as expressed by Equation (23), FIG. 22Billustrates the derivative of the applied temperature change. Thisderivative was calculated numerically from the measured temporal profileof FIG. 22A, then filtered numerically to simulate the 4-stage,24-dB/octave low-pass filter of the lock-in amplifier. Comparison to themeasured rotation error, reproduced in FIG. 22B, shows a reasonableagreement between the two curves, in agreement with Equation (23).

In a quadrupolar winding, as illustrated in FIG. 23, the first(outermost) layer is a portion of the sensing fiber located close to oneof the two Sagnac loop ends closest to the coupler (e.g., betweenpositions z=0 and z=L₁). The second layer, underneath it, is a portionof the sensing fiber located at the opposite end of the coil (L_(n-1)<z<L_(n)=L). The third layer is a portion of the sensing fiber locatednext to the second layer (L_(n-2)<z <L_(n-1)), etc. Just after the heathas been turned on, the first layer of the coil heats up first, and as aresult, the differential phase between the counter-propagating waveschanges (e.g., increases). As heat continues to be applied, itpropagates radially into the coil, and the second layer, then the thirdlayer, start to warm up. In a quadrupolar or a bipolar winding, thefirst and second layers are located symmetrically in the Sagnac loop.Hence as the second layer heats up, the thermal phase shift it inducesbegins to cancel that induced in the first layer. The same cancellationprocess takes place for the deeper layers. The total phase shift,however, continues to increase because the first layer heats up fasterthan the internal layers. Eventually, the temperature of the outer layerreaches some maximum value, and as more internal layers gradually heatup the total thermal phase decreases. If heat is applied long enough,the temperature along the fiber reaches a steady-state distribution, andthe thermal phase shift vanishes.

This behavior is consistent with the observed behavior of the thermallyinduced signal, which increases first, then decreases over time, asshown by FIG. 22B. The measured signal closely follows the temperaturederivative for about 1 second after the heat was turned on. For longertimes, the two curves disagree in that the measured rotation error curvedrops below the temperature derivative curve because, by then, heat hasreached deep into the coil and the quadrupolar winding starts cancelingthe thermal phase shift. Just after the heat is turned off (around t=5.5seconds in FIG. 22B), the rotation error becomes negative. The reason isthat at that time, the outermost layer starts to cool down. Hence thesign of the temperature gradient is reversed, and so is the sign of therotation error.

When the air-core fiber was replaced by the solid-core fiber coil, thebehavior of the gyroscope was similar, as shown by FIGS. 24A and 24B.The rotation error increased just after the start of the heat pulse,then decreased, and finally became negative after the heat was turnedoff. However, the solid-core fiber gyroscope was clearly much moresensitive to asymmetric heating than was the air-core FOG. Forcomparable applied peak derivative {dot over (T)} (75.5° C./s for theSMF28 vs. 41.1° C./s for the PBF), the error signal was about 10 timeslarger for the solid-core fiber gyroscope, as shown by a comparison ofFIGS. 22B and 24B.

The measurements provided the temperature derivative {dot over (T)}(z)at all times but only at the surface of the coil (z=0). It wasconsequently not possible to apply Equation (23) and extract from themeasured rotation error signals a value for the Shupe constant S of thetwo fiber coils. However, the thermal performance of the two gyroscopescan still be compared by making two observations. First, because the twocoils have identical diameter and thickness, the rates of heat flow areexpected to be comparable in the two coils. Second, based on the abovediscussion regarding the dynamic of heat flow in a quadrupolar coil, thetotal thermal phase shift is expected to reach its maximum shortly afterthe first layer has started to heat up. The maximum thermally inducedrotation error can therefore be approximated by:

$\begin{matrix}{\Omega_{E,\max} \approx {\frac{n^{2}S\; L}{D}{\int_{0}^{L_{1}/L}{\left( {1 - {2z^{\prime}}} \right){\overset{.}{T}\left( z^{\prime} \right)}{\mathbb{d}z^{\prime}}}}}} & (24)\end{matrix}$Furthermore, the rate of temperature change of the outermost layer isexpected to be close to the rate of temperature change measured at thesurface of the coil, and to be weakly dependent on z′. Hence, inEquation (24), {dot over (T)}(z′) can be taken out of the integral,which shows that Ω_(E,max) should scale approximately linearly with themeasured surface temperature derivative.

To verify this approximation, the dependence of the maximum rotationrate error on the applied temperature gradient was measured for eachgyroscope. For example, in the measurement shown by FIG. 22B, themaximum rotation rate error is equal to 0.02 deg/s, and at the time themaximum rotation rate error occurred (t≈1.8 s), the applied temperaturegradient was about 41.1° C./s. FIG. 25 shows the dependence of themaximum rotation rate error on the applied temperature gradient measuredin both the conventional solid-core fiber gyroscope and in the air-corefiber gyroscope. The maximum rotation rate error increases roughlylinearly with applied temperature gradient, which confirms the validityof the approximation. The slope of these dependencies are 2.4×10⁻³deg/s/(° C./s) for the SMF28 fiber gyroscope, and 2.9×10⁻⁴ deg/s/(°C./s) for the air-core fiber gyroscope. After correcting for theslightly different length L of the two sensing fibers, for identicalcoil lengths, the air-core fiber gyroscope is 6.5 times less sensitiveto temperature gradients than the conventional FOG.

Independent thermal measurements performed on short pieces of the samefibers showed that the ratio of Shupe constants for the SMF28 and theBlaze PBF is 3.6, as shown in Table 3. When using these values inEquation (23), together with mode effective indices of 0.99 and 1.44 forthe two fibers, respectively, and assuming identical coils, the air-corefiber is expected to be 7.6 times less sensitive than the solid-corefiber to thermal perturbations. This value is in good agreement with ourexperimental value of 6.5. The small difference (about 13%) may be dueto slightly different heat propagation properties in the coils, which isexpected since an air-core fiber constitutes a better thermal insulator.The measured value of 6.5 is also in excellent agreement with thetheoretically predicted ratio of 6.6 for these two fibers. In any case,measured and theoretical values demonstrate unequivocally thesignificant advantage of using an air-core fiber in a FOG to reduce itsthermal sensitivity. Further design improvements (e.g., optimization ofthe jacket) can result in the Shupe constant being reduced by anotherfactor of about 3, bringing the total improvement over a conventionalcoil to a factor of about 23.

The second significant long-term stability improvement provided by anair-core fiber gyroscope is a dramatic reduction in the non-reciprocalKerr effect. To illustrate this improvement, the magnitude of theKerr-induced drift in the air-core FOG was measured by observing thechange in the gyroscope output when the power between the twocounter-propagating signals was intentionally unbalanced. To be able toobserve this very weak effect, the IOC was replaced by a 10% fibercoupler, which provided a strong imbalance between thecounter-propagating powers. This change was accompanied by thereplacement of the other components present on the IOC (polarizationfilter and phase modulator) by a standard fiber polarizer and apiezoelectric fiber phase modulator. The SFS, which almost completelycancels the Kerr effect, was replaced by a narrow-band semiconductorlaser. With the narrow-band source, the noise due to coherentback-scattering from the fiber was quite large (about 19 dB higher thanwith the SFS). In fact, even with the largest input power tolerated bythe optical components (e.g., 50 mW), back-scattering noise exceeded theKerr-induced signal. In other words, the Kerr effect of the air-corefiber gyroscope was too weak to measure. Nevertheless, by recognizingthat the Kerr phase shift was at most equal to the(back-scattering-dominated) noise, this measurement provides an estimateof the upper bound value of the fiber mode's Kerr constant. Aftercorrecting for the known Kerr contribution from the (solid-core) fiberpigtails inside the Sagnac loop, the Kerr constant was found to bereduced by at least a factor of 50 compared to the same gyroscope usingthe SMF28-fiber coil. This result confirms that the Kerr effect issubstantially reduced in an air-core fiber, by a factor of 50 or more.

While the effect of a magnetic field on the air-core fiber gyroscope wasnot measured directly, the Verdet (Faraday) constant of a short lengthof the air-core fiber was measured. These measurements indicate that forequal length, the Faraday rotation induced by an applied magnetic fieldis about 160 times weaker in the air-core fiber than in an SMF28 fiber.Inferring an accurate value of the Verdet constant of the air-core fiberfrom this result utilizes a precise knowledge of the fiber'sbirefringence. This constant is estimated to be at least a factor ofabout 10 dB smaller than that of an SMF28 fiber, and it could be as lowas about 26 dB smaller than that of an SMF28 fiber. In practice, anair-core FOG requires much less μ-metal shielding (if any) than docurrent commercial FOGs, which will reduce the size, weight, and cost ofthe air-core FOG as compared to conventional FOGs.

In certain embodiments described herein, the temperature dependence ofan FOG is advantageously reduced by using an air-core fiber. In certainother embodiments, the temperature dependence of other types ofinterferometric fiber sensors can also be advantageously reduced. Suchfiber sensors include, but are not limited to, sensors based on opticalinterferometers such as Mach-Zehnder interferometers, Michelsoninterferometers, Fabry-Perot interferometers, ring interferometers,fiber Bragg gratings, long-period fiber Bragg gratings, and Fox-Smithinterferometers. In certain embodiments in which the fiber sensorutilizes a relatively short length of air-core fiber, the additionalcosts of such fibers are less of an issue in producing these improvedfiber sensors.

In certain embodiments described herein, the temperature dependence ofan FOG is advantageously reduced by using a Bragg fiber. In certainother embodiments, the temperature dependence of other types ofinterferometric fiber sensors can also be advantageously reduced. Suchfiber sensors include, but are not limited to, sensors based on opticalinterferometers such as Mach-Zehnder interferometers, Michelsoninterferometers, Fabry-Perot interferometers, ring interferometers,fiber Bragg gratings, long-period fiber Bragg gratings, and Fox-Smithinterferometers. In certain embodiments in which the fiber sensorutilizes a relatively short length of Bragg fiber, the additional costsof such fibers are less of an issue in producing these improved fibersensors.

While certain embodiments have been described herein as having aphotonic-bandgap fiber with a triangular pattern of holes in thecladding, other embodiments can utilize a photonic-bandgap fiber havingan arrangement of cladding holes that is different from triangular,provided that the fiber still supports a bandgap and the introduction ofa hollow core defect supports one or more core-guided modes localizedwithin this defect. For example, such conditions are satisfied by afiber with a cladding having a so-called Kagome lattice, as described in“Large-pitch kagome-structured hollow-core photonic crystal fiber,” byF. Couny et al., Optics Letters, Vol. 31, No. 34, pp. 3574-3576(December 2006). In certain other embodiments, a hollow-core fiber isutilized which do not exhibit a bandgap but still transmit lightconfined largely in the core over appreciable distances (e.g.,millimeters and larger).

Those skilled in the art will appreciate that the methods and designsdescribed above have additional applications and that the relevantapplications are not limited to those specifically recited above.Moreover, the present invention may be embodied in other specific formswithout departing from the essential characteristics as describedherein. The embodiments described above are to be considered in allrespects as illustrative only and not restrictive in any manner.

1. An optical sensor comprising: a directional coupler comprising atleast a first port, a second port, and a third port, the first port inoptical communication with the second port and with the third port suchthat a first optical signal received by the first port is split into asecond optical signal that propagates to the second port and a thirdoptical signal that propagates to the third port; and a photonic bandgapfiber having a hollow core and an inner cladding generally surroundingthe core, the photonic bandgap fiber in optical communication with thesecond port and with the third port, wherein the second optical signaland the third optical signal counterpropagate through the photonicbandgap fiber and return to the third port and the second port,respectively, wherein the photonic bandgap fiber has a phase thermalconstant S less than 8 parts-per-million per degree Celsius.
 2. Theoptical sensor of claim 1, wherein the phase thermal constant S is lessthan 6 part-per-million per degree Celsius.
 3. The optical sensor ofclaim 1, wherein the phase thermal constant S is less than 4part-per-million per degree Celsius.
 4. The optical sensor of claim 1,wherein the phase thermal constant S is less than 1.4 part-per-millionper degree Celsius.
 5. The optical sensor of claim 1, wherein thephotonic bandgap fiber further comprises an outer cladding generallysurrounding the inner cladding.
 6. The optical sensor of claim 5,wherein the photonic bandgap fiber further comprises a jacket generallysurrounding the outer cladding.
 7. The optical sensor of claim 6,wherein the jacket comprises a polyimide layer having a thickness lessthan or equal to 5 microns.
 8. The optical sensor of claim 7, whereinthe polyimide layer has a thickness of about 2.5 microns.
 9. The opticalsensor of claim 7, wherein the jacket further comprises an amorphouscarbon coating generally surrounded by the polyimide layer.
 10. Theoptical sensor of claim 1, wherein the inner cladding comprises amaterial having a first refractive index and a periodic array of regionshaving a second refractive index less than the first refractive index,the regions having a center-to-center distance of Λ.
 11. The opticalsensor of claim 1, wherein the photonic bandgap fiber is a single-modefiber.
 12. The optical sensor of claim 1, wherein the photonic bandgapfiber is a multi-mode fiber.
 13. The optical sensor of claim 1, whereinthe photonic bandgap fiber is a single polarization fiber.
 14. Theoptical sensor of claim 1, wherein the photonic bandgap fiber furthercomprises an outer cladding generally surrounding the inner cladding anda jacket generally surrounding the outer cladding, wherein the outercladding has an area A_(cl), a Young's modulus E_(cl), and a coefficientof thermal expansion α_(cl), wherein the jacket has an area A_(J), aYoung's modulus E_(J), and a coefficient of thermal expansion α_(J),wherein a quantity$\frac{A_{J}}{A_{cl}}\frac{E_{J}}{E_{cl}}\frac{\alpha_{J}}{\alpha_{cl}}$is less than or equal to 2.5.
 15. The optical sensor of claim 14,wherein the quantity is less than or equal to
 1. 16. The optical sensorof claim 1, further comprising an optical detector in opticalcommunication with the directional coupler to receive the second opticalsignal and the third optical signal after the second and third opticalsignals have traversed the photonic bandgap fiber.
 17. The opticalsensor of claim 1, wherein the core contains air.
 18. The optical sensorof claim 1, wherein the photonic bandgap fiber is a coil.
 19. Theoptical sensor of claim 18, wherein the coil is wound with quadrupolewinding.
 20. A method for sensing comprising: providing a light signal;propagating a first portion of the light signal in a first directionthrough a portion of a photonic bandgap fiber having a hollow core andan inner cladding generally surrounding the core, wherein the photonicbandgap fiber has a phase thermal constant S less than 8parts-per-million per degree Celsius; propagating a second portion ofthe light signal in a second direction through the photonic bandgapfiber, the second direction opposite to the first direction; opticallyinterfering the first and second portions of the light signal after thefirst and second portions of the light signal propagate through thephotonic bandgap fiber, thereby producing an optical interferencesignal; subjecting at least a portion of the photonic bandgap fiber to aperturbation; and measuring variations in the optical interferencesignal caused by the perturbation.
 21. The method of claim 20, whereinthe core contains air.
 22. The method of claim 20, wherein theperturbation comprises a rotation of the portion of the photonic bandgapfiber.
 23. The method of claim 20, wherein the perturbation comprises achange of pressure applied to the portion of the photonic bandgap fiber.24. The method of claim 20, wherein the perturbation comprises amovement of the portion of the photonic bandgap fiber.
 25. An opticalsystem comprising: a light source having an output that emits a firstoptical signal; a directional coupler comprising at least a first port,a second port and a third port, the first port in optical communicationwith the light source to receive the first optical signal emitted fromthe light source, the first port in optical communication with thesecond port and with the third port such that the first optical signalreceived by the first port is split into a second optical signal thatpropagates to the second port and a third optical signal that propagatesto the third port; a photonic bandgap fiber having a hollow core, aninner cladding generally surrounding the core, an outer claddinggenerally surrounding the inner cladding, and a jacket generallysurrounding the outer cladding, the photonic bandgap fiber in opticalcommunication with the second port and with the third port, wherein thesecond optical signal and the third optical signal counterpropagatethrough the photonic bandgap fiber and return to the third port and thesecond port, respectively, wherein the photonic bandgap fiber has aphase thermal constant S less than 8 ppm per degree Celsius; and anoptical detector in optical communication with the directional coupler,the optical detector receiving the counterpropagating second opticalsignal and the third optical signal after having traversed the photonicbandgap fiber.